Finite Sigma-Algebra Closure and Exact Measure Reduction
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
2. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [1], [0, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
3. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [0, 1], [2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
4. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [2], [0, 1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
5. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [0, 2], [1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
6. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [1, 2], [0, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
7. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
8. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [3], [0, 1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
9. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [0, 3], [1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
10. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [1, 3], [0, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
11. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [0, 1, 3], [2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
12. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [2, 3], [0, 1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
13. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [0, 2, 3], [1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
14. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [1, 2, 3], [0, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
15. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [0, 1, 2, 3], [4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
16. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [4], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
17. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [0, 4], [1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
18. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
19. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [0, 1, 4], [2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
20. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [2, 4], [0, 1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
21. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [0, 2, 4], [1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
22. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [1, 2, 4], [0, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
23. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
24. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [3, 4], [0, 1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
25. Every Complement and Binary Union on a Finite Carrier
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.
View answer and explanation
(7, [[], [0, 3, 4], [1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], Yes, Yes, [], [], Yes, Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-sigma-qualification · candidate
26. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
27. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [1], [0, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
28. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [0, 1], [2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
29. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [2], [0, 1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
30. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [0, 2], [1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
31. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [1, 2], [0, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
32. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
33. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [3], [0, 1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
34. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [0, 3], [1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
35. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [1, 3], [0, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
36. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [0, 1, 3], [2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
37. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [2, 3], [0, 1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
38. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [0, 2, 3], [1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
39. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [1, 2, 3], [0, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
40. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [0, 1, 2, 3], [4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
41. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [4], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
42. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [0, 4], [1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
43. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
44. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [0, 1, 4], [2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
45. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [2, 4], [0, 1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
46. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [0, 2, 4], [1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
47. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [1, 2, 4], [0, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
48. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
49. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [3, 4], [0, 1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
50. List Missing Closure Obligations without Guessing a Sigma-Algebra
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?).
View answer and explanation
(7, [[], [0, 3, 4], [1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-closure-inventory · candidate
51. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 1/3, 2/3, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
52. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [1], [0, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 2/5, 3/5, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
53. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [0, 1], [2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 3/7, 4/7, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
54. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [2], [0, 1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 4/9, 5/9, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
55. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [0, 2], [1, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 5/11, 6/11, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
56. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [1, 2], [0, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 6/13, 7/13, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
57. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 7/15, 8/15, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
58. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [3], [0, 1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 8/17, 9/17, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
59. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [0, 3], [1, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 9/19, 10/19, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
60. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [1, 3], [0, 2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 10/21, 11/21, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
61. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [0, 1, 3], [2, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 11/23, 12/23, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
62. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [2, 3], [0, 1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 12/25, 13/25, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
63. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [0, 2, 3], [1, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 13/27, 14/27, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
64. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [1, 2, 3], [0, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 14/29, 15/29, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
65. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [0, 1, 2, 3], [4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 15/31, 16/31, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
66. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [4], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 16/33, 17/33, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
67. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [0, 4], [1, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 17/35, 18/35, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
68. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 18/37, 19/37, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
69. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [0, 1, 4], [2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 19/39, 20/39, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
70. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [2, 4], [0, 1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 20/41, 21/41, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
71. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [0, 2, 4], [1, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 21/43, 22/43, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
72. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [1, 2, 4], [0, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 22/45, 23/45, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
73. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 23/47, 24/47, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
74. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [3, 4], [0, 1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 24/49, 25/49, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
75. Empty-Set Zero and Every Disjoint-Pair Additivity Check
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.
View answer and explanation
(7, [[], [0, 3, 4], [1, 2, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 25/51, 26/51, 1], Yes, [], Yes)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-measure-additivity-audit · candidate
76. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
77. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
78. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
79. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
80. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
81. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
82. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
83. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
84. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
85. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
86. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
87. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
88. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
89. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
90. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
91. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
92. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
93. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
94. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
95. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
96. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
97. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
98. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
99. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
100. A Finite Measure Need Not Have Total Mass One
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.
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate
Canonical question JSONL · English JSONL · Validation report