{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0001","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d4d16ddb68587f6b1a831bc6cd794715de482c21887b7614588dbfbb2263123","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235001,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0],[1,2,3,4,5,6]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0002","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c97fe471f524e018c3e0dc5b15d301f6b6c3b2fd6eeaf4393a7ca0cd85e9f7b9","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235002,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,3,4,5,6],[1]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1], [0, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [1], [0, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1], [0, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0003","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc2876307b051dbad684644d78d13a0ec76ef6025be6b5356bb07b086b364e48","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235003,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,1],[2,3,4,5,6]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1], [2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [0, 1], [2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1], [2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0004","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7dd00221937e63831a84ef63a5819fadeb441778a102215a027e56f21c4ee88","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235004,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,3,4,5,6],[2]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [2], [0, 1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [2], [0, 1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [2], [0, 1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0005","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a41692acb2f2e47686e6829cf9faccbec80af928057dc8dba4e55498cbe7ab9","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235005,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,2],[1,3,4,5,6]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 2], [1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [0, 2], [1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 2], [1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0006","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca4326da8f23256c34b5f2fefd30ff461c58d5e671e7dc1757587079b622aa47","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235006,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,3,4,5,6],[1,2]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 2], [0, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [1, 2], [0, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 2], [0, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0007","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afeaa0957aa10fb2551348d0acc1717e42d8993f1c3907da487900c2b763f159","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235007,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2],[3,4,5,6]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0008","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"318a8756fd51765a761f89decc32ee494e2384fb697ea9566baf20fba3d9e997","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235008,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,4,5,6],[3]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [3], [0, 1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [3], [0, 1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [3], [0, 1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0009","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e9cd1f028fcc92e756bd64811e0ae8651098629cea40cc25ffed076e959f8cca","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235009,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,3],[1,2,4,5,6]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 3], [1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [0, 3], [1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 3], [1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0010","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa82fd177fb13d04a2c3399eee7120fa87ca478e57e6c5b61df644e258ad6e17","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235010,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,4,5,6],[1,3]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 3], [0, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [1, 3], [0, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 3], [0, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0011","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b9981719129b287d24425601c6b02a5f0766c15ed22e8be8dc1390257807f2b","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235011,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,3],[2,4,5,6]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 3], [2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [0, 1, 3], [2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 3], [2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0012","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"046654adb0fa89edfebd1b9d95b4758e0318c40dee8888bd7b2048810adf5d7e","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235012,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,4,5,6],[2,3]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [2, 3], [0, 1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [2, 3], [0, 1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [2, 3], [0, 1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0013","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2071055778666eafd409686e554bed96a0d2cedec20726d421aa10f319a26fe0","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235013,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,3],[1,4,5,6]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 2, 3], [1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [0, 2, 3], [1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 2, 3], [1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0014","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40a3b6e310951d58d4bc61d4cdfc2534452cda60d9983da937deaf894f47d6b9","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235014,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,4,5,6],[1,2,3]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 2, 3], [0, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [1, 2, 3], [0, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 2, 3], [0, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0015","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bc396aaf9e03115c1996065d3ebde5a5770879be02526933b01b550249ecf15","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235015,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,3],[4,5,6]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2, 3], [4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [0, 1, 2, 3], [4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 2, 3], [4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0016","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6d142cf6e1c4ec321ac32933a99227da77d07191e64c5c384d2340c7d5b3f0b","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235016,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,3,5,6],[4]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [4], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [4], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [4], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0017","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0da0a6f8c127f8d9a7952d4d69dd1180d4e3c079937e419279dc8cb642c0f43d","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235017,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,4],[1,2,3,5,6]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 4], [1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [0, 4], [1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 4], [1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0018","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"065e473820a27d972d29e88223ad032575191b0188edfa412699cce93d690cf8","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235018,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,3,5,6],[1,4]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0019","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29aae32d4c6548328ab35ec73e4cf6d21d9dc2c645113aac0a3dbb11e8f78db8","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235019,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,4],[2,3,5,6]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 4], [2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [0, 1, 4], [2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 4], [2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0020","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b93b61c01fefc8fcc2bb056ac818395948b7126b147a66e3af9948544296af47","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235020,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,3,5,6],[2,4]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [2, 4], [0, 1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [2, 4], [0, 1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [2, 4], [0, 1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0021","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af2c1690df57dbdadf78db61b7cd75804076c5dcfad4f4130bb2be7b001e1dba","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235021,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,4],[1,3,5,6]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 2, 4], [1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [0, 2, 4], [1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 2, 4], [1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0022","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0080a350f6e9a38e6cb7541fa16647145176b05f066f9ad7ab6ff650327f1ee9","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235022,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,3,5,6],[1,2,4]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 2, 4], [0, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [1, 2, 4], [0, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 2, 4], [0, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0023","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16043c1c97415a59597a34d56929595e80e14ab147496988c15b70129b39432d","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235023,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,4],[3,5,6]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0024","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a603a01f5d1fe130ea91d7c93f97893ca662259d929d9be63beeb37c538e2a5c","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235024,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,5,6],[3,4]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [3, 4], [0, 1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [3, 4], [0, 1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [3, 4], [0, 1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0025","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2211d9eeadae7182538c2446821caa3fe1b2d2f8ec7cfb1e3a986ec636910a1","family_id":"qf-mod23-measure-sigma-qualification","seed":202610235025,"semantic_template":"qf-mod23-measure-sigma-qualification/{\"family\":[[],[0,1,2,3,4,5,6],[0,3,4],[1,2,5,6]],\"n\":7}"},"fields":{"title":"Every Complement and Binary Union on a Finite Carrier","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 3, 4], [1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.","answer":{"kind":"expression","value":"(7, [[], [0, 3, 4], [1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)","explanation":"A finite carrier permits countable unions to reduce to finitely many distinct sets. This reduction still requires the empty/full and all complement/union checks. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 3, 4], [1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)."},"skills":["Every Complement and Binary Union on a Finite Carrier"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0026","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c608587168ca8a248ffda28c3099aaabc5f8923ed638df5326042fd77ca499f","family_id":"qf-mod23-measure-closure-inventory","seed":202610235026,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0],[1,2,3,4,5,6]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0027","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcb20145ce1ad52627ba457ac78e2411ad332b9c4ea3adc57bf8f75165f532bc","family_id":"qf-mod23-measure-closure-inventory","seed":202610235027,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,3,4,5,6],[1]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1], [0, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [1], [0, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1], [0, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0028","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7ef64f8a88c30bd7b517c38eeaaae6ddb01f094a0cdd943989808e7e7d75513","family_id":"qf-mod23-measure-closure-inventory","seed":202610235028,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,1],[2,3,4,5,6]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1], [2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [0, 1], [2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1], [2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0029","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c6997cdf348f32d1f98fe023877fdb5391911568f8b510061174a04f7fa2574","family_id":"qf-mod23-measure-closure-inventory","seed":202610235029,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,3,4,5,6],[2]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [2], [0, 1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [2], [0, 1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [2], [0, 1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0030","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a20f84671bf04939a46d31035afa66f3da6eca8f7bf938931c9a52a223ae376","family_id":"qf-mod23-measure-closure-inventory","seed":202610235030,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,2],[1,3,4,5,6]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 2], [1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [0, 2], [1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 2], [1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0031","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6764e50cf091459c58133aa268cc3000fbc21277c2d477572f69c5261056f22a","family_id":"qf-mod23-measure-closure-inventory","seed":202610235031,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,3,4,5,6],[1,2]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 2], [0, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [1, 2], [0, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 2], [0, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0032","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06abcb29dd57246e6274f22cec789fa7b18906954dea25367cd0b1f04e2786f8","family_id":"qf-mod23-measure-closure-inventory","seed":202610235032,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2],[3,4,5,6]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0033","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9e9e99f9008bd24467fbb2f2e3096645c52eabf514dbf9b892d1df6d2d93bb8","family_id":"qf-mod23-measure-closure-inventory","seed":202610235033,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,4,5,6],[3]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [3], [0, 1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [3], [0, 1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [3], [0, 1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0034","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e44f5507e27f6b6533e2d204bca6a1017b011d699e4e5b04ccb29df4508f5b56","family_id":"qf-mod23-measure-closure-inventory","seed":202610235034,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,3],[1,2,4,5,6]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 3], [1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [0, 3], [1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 3], [1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0035","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0089e391798c30ad6eafb3901830c70cabf87a9db9071bba62af63239af950d9","family_id":"qf-mod23-measure-closure-inventory","seed":202610235035,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,4,5,6],[1,3]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 3], [0, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [1, 3], [0, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 3], [0, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0036","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2fdc090d82512a99c75bee8cefa9ef279dcebc455d5ce9d1f8df8217252fc5c","family_id":"qf-mod23-measure-closure-inventory","seed":202610235036,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,3],[2,4,5,6]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 3], [2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [0, 1, 3], [2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 3], [2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0037","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db06cc7a701276ecf3df6d16b7cef63bcf171ec77adeec07c05913ec80c8c401","family_id":"qf-mod23-measure-closure-inventory","seed":202610235037,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,4,5,6],[2,3]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [2, 3], [0, 1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [2, 3], [0, 1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [2, 3], [0, 1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0038","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc7514d1c81ec8ffc2f91e290c5aeb36f6b8170e9a1c500da9b5e55fb91dd0a4","family_id":"qf-mod23-measure-closure-inventory","seed":202610235038,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,3],[1,4,5,6]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 2, 3], [1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [0, 2, 3], [1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 2, 3], [1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0039","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"709635f3778064ef4d25dad1070d5fe89cff77a09dd7ccc11087aeb2a68561b6","family_id":"qf-mod23-measure-closure-inventory","seed":202610235039,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,4,5,6],[1,2,3]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 2, 3], [0, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [1, 2, 3], [0, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 2, 3], [0, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0040","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9354899ba987b93dec7ce62884743bb4cf772629809885da4dbd94a001e8def1","family_id":"qf-mod23-measure-closure-inventory","seed":202610235040,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,3],[4,5,6]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2, 3], [4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [0, 1, 2, 3], [4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 2, 3], [4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0041","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"335a9de2f588e5f395ee695c563ee58a8bbc92d3a7887e9f5f4b45c739c5c74d","family_id":"qf-mod23-measure-closure-inventory","seed":202610235041,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,3,5,6],[4]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [4], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [4], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [4], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0042","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e09fe5c66d325060b74144ad9fbed3cd0490d82d725fe44b88f0115839038801","family_id":"qf-mod23-measure-closure-inventory","seed":202610235042,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,4],[1,2,3,5,6]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 4], [1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [0, 4], [1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 4], [1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0043","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9c29d546165f1956c132a7188353944487978a52bcf71e8a025dba0a0136d03","family_id":"qf-mod23-measure-closure-inventory","seed":202610235043,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,3,5,6],[1,4]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0044","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35ee2cdba3d855f4f276d9ccdc2dc933022f4c4d7e6e208aaae1884da46efe4a","family_id":"qf-mod23-measure-closure-inventory","seed":202610235044,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,4],[2,3,5,6]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 4], [2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [0, 1, 4], [2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 4], [2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0045","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb4ead12415c4751e74d03490140193a9aac3e64dde985ac81a124b760eac13f","family_id":"qf-mod23-measure-closure-inventory","seed":202610235045,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,3,5,6],[2,4]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [2, 4], [0, 1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [2, 4], [0, 1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [2, 4], [0, 1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0046","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aabfad130590ede261d1d8c5af0f963ed2ac5b3e59a5722e21d67dda8d2389b8","family_id":"qf-mod23-measure-closure-inventory","seed":202610235046,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,4],[1,3,5,6]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 2, 4], [1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [0, 2, 4], [1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 2, 4], [1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0047","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0053e3c104dd3e6769a90d3c7bfb0e1cf5e6be9b2c6ac9de2efbd4bfcfa1d5ba","family_id":"qf-mod23-measure-closure-inventory","seed":202610235047,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,3,5,6],[1,2,4]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 2, 4], [0, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [1, 2, 4], [0, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 2, 4], [0, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0048","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff76c04964d8121c46629c50b8f05abbe29dc1419ecba8bcf35a0181631e2dfc","family_id":"qf-mod23-measure-closure-inventory","seed":202610235048,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,4],[3,5,6]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0049","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cb2f713da6c42ea4b51afa25e3e3481053af7587db7eadd9ba22c04d6d3d04e","family_id":"qf-mod23-measure-closure-inventory","seed":202610235049,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,5,6],[3,4]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [3, 4], [0, 1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [3, 4], [0, 1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [3, 4], [0, 1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0050","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0002886c686c92545cdae7de83e5483ca30ab094bcbbb8f689c95b7da6f9683","family_id":"qf-mod23-measure-closure-inventory","seed":202610235050,"semantic_template":"qf-mod23-measure-closure-inventory/{\"family\":[[],[0,1,2,3,4,5,6],[0,3,4],[1,2,5,6]],\"n\":7}"},"fields":{"title":"List Missing Closure Obligations without Guessing a Sigma-Algebra","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 3, 4], [1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).","answer":{"kind":"expression","value":"(7, [[], [0, 3, 4], [1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)","explanation":"Enumerate every declared family member and ordered pair. The finite-carrier reduction is not a general infinite-carrier closure theorem. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 3, 4], [1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)."},"skills":["List Missing Closure Obligations without Guessing a Sigma-Algebra"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0051","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd50017ec0ca60f7f3ccbcdd0ea441da3b982a6104a6ca51148be3205d899e21","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235051,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0],[1,2,3,4,5,6]],\"n\":7,\"values\":[[0,1],[1,3],[2,3],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 1/3, 2/3, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 1/3, 2/3, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 1/3, 2/3, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0052","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eaa5150e309b81ee7775c79ea441aa74d8fbf3ca33c7f0e93ebaa0c17cfb39c3","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235052,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,3,4,5,6],[1]],\"n\":7,\"values\":[[0,1],[2,5],[3,5],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1], [0, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 2/5, 3/5, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [1], [0, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 2/5, 3/5, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1], [0, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 2/5, 3/5, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0053","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bff0e1b1ba230e5a0ef6179f9452420b2bb04bef51a31bd3a2922022e185486","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235053,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1],[2,3,4,5,6]],\"n\":7,\"values\":[[0,1],[3,7],[4,7],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1], [2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 3/7, 4/7, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 1], [2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 3/7, 4/7, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1], [2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 3/7, 4/7, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0054","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3dab24ba8a12b993246e59684eeafa44043bacc51a4b9dbd6779a6ce8c68762","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235054,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,3,4,5,6],[2]],\"n\":7,\"values\":[[0,1],[4,9],[5,9],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [2], [0, 1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 4/9, 5/9, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [2], [0, 1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 4/9, 5/9, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [2], [0, 1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 4/9, 5/9, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0055","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28b9c3d85c5b504a3159bb56d50e97684b7eda37b598528f5cdd77feb18f8a7b","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235055,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,2],[1,3,4,5,6]],\"n\":7,\"values\":[[0,1],[5,11],[6,11],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 2], [1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 5/11, 6/11, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 2], [1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 5/11, 6/11, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 2], [1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 5/11, 6/11, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0056","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2de11fc9f6acd2a1ea18fafecee48697f3c9c13c8139a3df2d73e4f2d1161d7d","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235056,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,3,4,5,6],[1,2]],\"n\":7,\"values\":[[0,1],[6,13],[7,13],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 2], [0, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 6/13, 7/13, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [1, 2], [0, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 6/13, 7/13, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 2], [0, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 6/13, 7/13, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0057","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abdf1a86261e3ef968fdda4fae0eaee45a503c83cd0d9af839cc770546b4cb07","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235057,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2],[3,4,5,6]],\"n\":7,\"values\":[[0,1],[7,15],[8,15],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 7/15, 8/15, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 7/15, 8/15, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 7/15, 8/15, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0058","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b096d450030c18389949845e56f06723854e098d210c1dc9054f37e578eae172","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235058,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,4,5,6],[3]],\"n\":7,\"values\":[[0,1],[8,17],[9,17],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [3], [0, 1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 8/17, 9/17, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [3], [0, 1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 8/17, 9/17, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [3], [0, 1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 8/17, 9/17, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0059","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d799c4c49b94ae97c6841b1db0a1a9250b0fce13c9401dceac664dcdff0942a","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235059,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,3],[1,2,4,5,6]],\"n\":7,\"values\":[[0,1],[9,19],[10,19],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 3], [1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 9/19, 10/19, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 3], [1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 9/19, 10/19, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 3], [1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 9/19, 10/19, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0060","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc39439197fdcd4fe0b573caccba7a8c9169e2f0915ec1de99c83c98a8255737","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235060,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,4,5,6],[1,3]],\"n\":7,\"values\":[[0,1],[10,21],[11,21],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 3], [0, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 10/21, 11/21, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [1, 3], [0, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 10/21, 11/21, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 3], [0, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 10/21, 11/21, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0061","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"340b5959063db14334f309a151edf40249022058b59178d549c8779112013e55","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235061,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,3],[2,4,5,6]],\"n\":7,\"values\":[[0,1],[11,23],[12,23],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 3], [2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 11/23, 12/23, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 1, 3], [2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 11/23, 12/23, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 3], [2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 11/23, 12/23, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0062","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ed00889d8f6b0c93f44c5b546f7d0a4bf76ac71f1588c15d30d53fa116afbf6","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235062,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,4,5,6],[2,3]],\"n\":7,\"values\":[[0,1],[12,25],[13,25],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [2, 3], [0, 1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 12/25, 13/25, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [2, 3], [0, 1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 12/25, 13/25, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [2, 3], [0, 1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 12/25, 13/25, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0063","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63278b60b753a597b5c3296f27f42cd9a1cb7db7f92b67ca548748da13073402","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235063,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,3],[1,4,5,6]],\"n\":7,\"values\":[[0,1],[13,27],[14,27],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 2, 3], [1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 13/27, 14/27, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 2, 3], [1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 13/27, 14/27, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 2, 3], [1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 13/27, 14/27, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0064","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ce8051ef54f9b010ba0f90e2d6949f382078c32cdaf11f3f75dba54345f5b75","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235064,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,4,5,6],[1,2,3]],\"n\":7,\"values\":[[0,1],[14,29],[15,29],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 2, 3], [0, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 14/29, 15/29, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [1, 2, 3], [0, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 14/29, 15/29, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 2, 3], [0, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 14/29, 15/29, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0065","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70f87f84ee401c856a9f1d2c82f2128035582ef19c061c58d156f124d2f679ed","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235065,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,3],[4,5,6]],\"n\":7,\"values\":[[0,1],[15,31],[16,31],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2, 3], [4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 15/31, 16/31, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 1, 2, 3], [4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 15/31, 16/31, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 2, 3], [4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 15/31, 16/31, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0066","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"677089e73536e6c2e7c3ada360c899a8435602eaa2e8f9a041575b9e907772c5","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235066,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,3,5,6],[4]],\"n\":7,\"values\":[[0,1],[16,33],[17,33],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [4], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 16/33, 17/33, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [4], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 16/33, 17/33, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [4], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 16/33, 17/33, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0067","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ccf50bebc585fe0ddb8643d3a5751444bcb99da07a4453dc2910236fc75bc74","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235067,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,4],[1,2,3,5,6]],\"n\":7,\"values\":[[0,1],[17,35],[18,35],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 4], [1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 17/35, 18/35, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 4], [1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 17/35, 18/35, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 4], [1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 17/35, 18/35, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0068","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2391749976c5f8043d5dad2fc4c30592436b700181bdce47849320fd5b944d8","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235068,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,3,5,6],[1,4]],\"n\":7,\"values\":[[0,1],[18,37],[19,37],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 18/37, 19/37, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 18/37, 19/37, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 18/37, 19/37, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0069","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe793e6b2fb31c3d71c44ef59aa8f9a11fa77826cc9cd1d16f724e0c228b504c","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235069,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,4],[2,3,5,6]],\"n\":7,\"values\":[[0,1],[19,39],[20,39],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 4], [2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 19/39, 20/39, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 1, 4], [2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 19/39, 20/39, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 4], [2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 19/39, 20/39, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0070","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d81cd1d8404131abf0c06a493e9abc88c37dc562b7314ee0bc777aa197d3dae","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235070,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,3,5,6],[2,4]],\"n\":7,\"values\":[[0,1],[20,41],[21,41],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [2, 4], [0, 1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 20/41, 21/41, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [2, 4], [0, 1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 20/41, 21/41, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [2, 4], [0, 1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 20/41, 21/41, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0071","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd60096f0a591ec0fcfc090c70a9c3454256bd9f31ea219eab816655ac61e606","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235071,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,4],[1,3,5,6]],\"n\":7,\"values\":[[0,1],[21,43],[22,43],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 2, 4], [1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 21/43, 22/43, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 2, 4], [1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 21/43, 22/43, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 2, 4], [1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 21/43, 22/43, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0072","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5801d72d056bc9b855355fcf40dcc47550db56fe97a834691ea8b10097745a2e","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235072,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,3,5,6],[1,2,4]],\"n\":7,\"values\":[[0,1],[22,45],[23,45],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 2, 4], [0, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 22/45, 23/45, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [1, 2, 4], [0, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 22/45, 23/45, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 2, 4], [0, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 22/45, 23/45, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0073","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67924c1adf43151fec2be53e493c280a41f4bc31f09a7398a4b458524efef695","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235073,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,4],[3,5,6]],\"n\":7,\"values\":[[0,1],[23,47],[24,47],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 23/47, 24/47, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 23/47, 24/47, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 23/47, 24/47, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0074","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35808d418a29146e848632f34eaaa5b3d5d4a6713d6a2e016eb449de4ec43c5b","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235074,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,5,6],[3,4]],\"n\":7,\"values\":[[0,1],[24,49],[25,49],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [3, 4], [0, 1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 24/49, 25/49, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [3, 4], [0, 1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 24/49, 25/49, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [3, 4], [0, 1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 24/49, 25/49, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0075","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8738a0b1de47036a3150ba69ca1e2d77bd39d11b5b2f83b147af9ec0ee57c14a","family_id":"qf-mod23-measure-measure-additivity-audit","seed":202610235075,"semantic_template":"qf-mod23-measure-measure-additivity-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,3,4],[1,2,5,6]],\"n\":7,\"values\":[[0,1],[25,51],[26,51],[1,1]]}"},"fields":{"title":"Empty-Set Zero and Every Disjoint-Pair Additivity Check","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 3, 4], [1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra and aligned nonnegative rational values=[0/1, 25/51, 26/51, 1/1]. Give (size,family,normalized values,empty set has measure zero?,every bad [A,B,measure of union,sum of measures] record for disjoint ordered pairs,finite countable-additivity reduction?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 3, 4], [1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 25/51, 26/51, 1], Yes, [], Yes)","explanation":"Include disjoint pairs involving the empty set. On this finite carrier there are only finitely many disjoint nonempty sets; zero empty mass and all finite disjoint-pair identities close this specified countable reduction. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 3, 4], [1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 25/51, 26/51, 1], Yes, [], Yes)."},"skills":["Empty-Set Zero and Every Disjoint-Pair Additivity Check"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0076","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85fdf8bca6cf4bf6ff1f67e8e5da727e7f4363b0ee6d51a6c425d4496b13060d","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235076,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0],[1,2,3,4,5,6]],\"n\":7,\"values\":[[0,1],[1,3],[2,3],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 1/3, 2/3, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 1/3, 2/3, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 1/3, 2/3, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0077","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0b11d341f01b3f9d6687dc0cda7e9184fa7416353eb37a0dfe59da5fbb4ed52","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235077,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,3,4,5,6],[1]],\"n\":7,\"values\":[[0,1],[2,5],[3,5],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1], [0, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 2/5, 3/5, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [1], [0, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 2/5, 3/5, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1], [0, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 2/5, 3/5, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0078","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21968fc2b23d071b2ba95bb6ceeb4119402285599effe38db6089e4c2b7b75a2","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235078,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1],[2,3,4,5,6]],\"n\":7,\"values\":[[0,1],[3,7],[4,7],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1], [2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 3/7, 4/7, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 1], [2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 3/7, 4/7, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1], [2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 3/7, 4/7, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0079","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b85f64b4293bf8de88470c12581c8a98433cda8fb37dbc8bbad3576ea3beface","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235079,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,3,4,5,6],[2]],\"n\":7,\"values\":[[0,1],[4,9],[5,9],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [2], [0, 1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 4/9, 5/9, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [2], [0, 1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 4/9, 5/9, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [2], [0, 1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 4/9, 5/9, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0080","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e13b2bee1ab72e37e0a7116115877b8ba916518fae2a9d49e405d4bdf95d780","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235080,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,2],[1,3,4,5,6]],\"n\":7,\"values\":[[0,1],[5,11],[6,11],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 2], [1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 5/11, 6/11, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 2], [1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 5/11, 6/11, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 2], [1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 5/11, 6/11, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0081","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cdaac39d53a17dcae1db2a0fb173ac2fb882ecccb27a2cdd7e533195e47daf7","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235081,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,3,4,5,6],[1,2]],\"n\":7,\"values\":[[0,1],[6,13],[7,13],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 2], [0, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 6/13, 7/13, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [1, 2], [0, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 6/13, 7/13, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 2], [0, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 6/13, 7/13, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0082","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a1df2f92c4265b8bce51bf6b9d3b8c1e0f039a22d9a94ba80f06da83ea13340","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235082,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2],[3,4,5,6]],\"n\":7,\"values\":[[0,1],[7,15],[8,15],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 7/15, 8/15, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 7/15, 8/15, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 7/15, 8/15, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0083","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6f3db22ea93c8165d550698764cb535921de2e56d3cbefd32ea0b33c94bdb4c","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235083,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,4,5,6],[3]],\"n\":7,\"values\":[[0,1],[8,17],[9,17],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [3], [0, 1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 8/17, 9/17, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [3], [0, 1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 8/17, 9/17, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [3], [0, 1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 8/17, 9/17, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0084","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"725b851befb89a9b93470c87ac5dc839dc14d5a8653aa0a62f109e01eba3d0db","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235084,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,3],[1,2,4,5,6]],\"n\":7,\"values\":[[0,1],[9,19],[10,19],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 3], [1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 9/19, 10/19, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 3], [1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 9/19, 10/19, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 3], [1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 9/19, 10/19, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0085","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"004b9e5000e5d213ec76c4e4bab18cc63bb24fc10efd5e450d97388c26c1c7f5","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235085,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,4,5,6],[1,3]],\"n\":7,\"values\":[[0,1],[10,21],[11,21],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 3], [0, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 10/21, 11/21, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [1, 3], [0, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 10/21, 11/21, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 3], [0, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 10/21, 11/21, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0086","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3ebc906705a3dbd87d15ebffb96c5742bc749d03a9cb553a5fb9afbf44c14ba","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235086,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,3],[2,4,5,6]],\"n\":7,\"values\":[[0,1],[11,23],[12,23],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 3], [2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 11/23, 12/23, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 1, 3], [2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 11/23, 12/23, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 3], [2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 11/23, 12/23, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0087","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3020afe4e47d6f2a579dab5cfbba04e67077f09141b2504b4954089c3ff83383","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235087,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,4,5,6],[2,3]],\"n\":7,\"values\":[[0,1],[12,25],[13,25],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [2, 3], [0, 1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 12/25, 13/25, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [2, 3], [0, 1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 12/25, 13/25, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [2, 3], [0, 1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 12/25, 13/25, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0088","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf317f57fefabfa0302d44cdd360d1584f8ce73ff836936abb9ea7695b25704f","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235088,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,3],[1,4,5,6]],\"n\":7,\"values\":[[0,1],[13,27],[14,27],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 2, 3], [1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 13/27, 14/27, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 2, 3], [1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 13/27, 14/27, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 2, 3], [1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 13/27, 14/27, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0089","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bc74290ca26c2e6f9cf029a08d8ab33a9db36802752243f352d82f16db6c9f5","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235089,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,4,5,6],[1,2,3]],\"n\":7,\"values\":[[0,1],[14,29],[15,29],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 2, 3], [0, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 14/29, 15/29, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [1, 2, 3], [0, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 14/29, 15/29, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 2, 3], [0, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 14/29, 15/29, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0090","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e0f428f31490cfeef83e4db2c4e19d3047fb94ac9211d117ce610a754ce52d6","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235090,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,3],[4,5,6]],\"n\":7,\"values\":[[0,1],[15,31],[16,31],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2, 3], [4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 15/31, 16/31, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 1, 2, 3], [4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 15/31, 16/31, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 2, 3], [4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 15/31, 16/31, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0091","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a6a19e387e2d5872aff58c4c68fe6a7040459e32a785245c597113401fab0d1","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235091,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,3,5,6],[4]],\"n\":7,\"values\":[[0,1],[16,33],[17,33],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [4], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 16/33, 17/33, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [4], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 16/33, 17/33, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [4], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 16/33, 17/33, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0092","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7632743014c54488c60154f07c489a2c52285a43b922a71db3a5b2f6cb45aba9","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235092,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,4],[1,2,3,5,6]],\"n\":7,\"values\":[[0,1],[17,35],[18,35],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 4], [1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 17/35, 18/35, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 4], [1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 17/35, 18/35, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 4], [1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 17/35, 18/35, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0093","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdfef1d5150c8e230def8576e26f10c7e03c5ae8fa32526e66aa62b34aa03772","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235093,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,3,5,6],[1,4]],\"n\":7,\"values\":[[0,1],[18,37],[19,37],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 18/37, 19/37, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 18/37, 19/37, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 18/37, 19/37, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0094","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a9fe9251a6bf0484f41c2baf9d9279ae24c7caf0be834346e301cc69c03d3cf","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235094,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,4],[2,3,5,6]],\"n\":7,\"values\":[[0,1],[19,39],[20,39],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 4], [2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 19/39, 20/39, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 1, 4], [2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 19/39, 20/39, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 4], [2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 19/39, 20/39, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0095","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6edf4f3d677be9a63e3a90e03b5934cfad8b9261e6d70211f84ef55426a11a4","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235095,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,3,5,6],[2,4]],\"n\":7,\"values\":[[0,1],[20,41],[21,41],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [2, 4], [0, 1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 20/41, 21/41, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [2, 4], [0, 1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 20/41, 21/41, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [2, 4], [0, 1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 20/41, 21/41, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0096","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a8ba1ec417f078afe0b599b2e5b18312c744f84af33ee1b7fbfe5e2d3f08844","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235096,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,2,4],[1,3,5,6]],\"n\":7,\"values\":[[0,1],[21,43],[22,43],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 2, 4], [1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 21/43, 22/43, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 2, 4], [1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 21/43, 22/43, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 2, 4], [1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 21/43, 22/43, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0097","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22f49237cde2e997f990a3e989682047de1c9b1142e3680feeb305faa3455707","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235097,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,3,5,6],[1,2,4]],\"n\":7,\"values\":[[0,1],[22,45],[23,45],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [1, 2, 4], [0, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 22/45, 23/45, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [1, 2, 4], [0, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 22/45, 23/45, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [1, 2, 4], [0, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 22/45, 23/45, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0098","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8ce70b99f6733995f9157c5e6404016ecd9251abcfd047e2cfe665f8b8936fc9","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235098,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,4],[3,5,6]],\"n\":7,\"values\":[[0,1],[23,47],[24,47],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 23/47, 24/47, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 23/47, 24/47, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 23/47, 24/47, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0099","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0784daf80c1117e004759b08fa1c9ff3dda41f8e2d32f0f6dcbce3e05915e68","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235099,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,1,2,5,6],[3,4]],\"n\":7,\"values\":[[0,1],[24,49],[25,49],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [3, 4], [0, 1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 24/49, 25/49, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [3, 4], [0, 1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 24/49, 25/49, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [3, 4], [0, 1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 24/49, 25/49, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0100","batch_id":"batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"810df586657199151f57db0471bd0f3b26e55e59a5bba2a7247ce1cc09a8a49b","family_id":"qf-mod23-measure-probability-measure-audit","seed":202610235100,"semantic_template":"qf-mod23-measure-probability-measure-audit/{\"family\":[[],[0,1,2,3,4,5,6],[0,3,4],[1,2,5,6]],\"n\":7,\"values\":[[0,1],[25,51],[26,51],[1,1]]}"},"fields":{"title":"A Finite Measure Need Not Have Total Mass One","stem":"Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 3, 4], [1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Require a qualified sigma-algebra; aligned nonnegative rational values=[0/1, 25/51, 26/51, 1/1]. Give (size,family,values,empty-zero?,all bad disjoint-pair records,total carrier mass,finite countable-additivity reduction?,probability measure?). Use exact reduced rational outputs with positive denominators, omitting /1 for integers.","answer":{"kind":"expression","value":"(7, [[], [0, 3, 4], [1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 25/51, 26/51, 1], Yes, [], 1, Yes, Yes)","explanation":"First check empty-zero and every disjoint pair on the finite carrier. A probability measure additionally requires total mass exactly 1; a valid measure of total 2 is not a probability measure. This explicit finite/exact slice does not automatically become a general infinite conclusion. The exact result, retaining the requested order, is (7, [[], [0, 3, 4], [1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 25/51, 26/51, 1], Yes, [], 1, Yes, Yes)."},"skills":["A Finite Measure Need Not Have Total Mass One"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}