# Question 10418: List Missing Closure Obligations without Guessing a Sigma-Algebra [Return to catalogue page 53](https://ocme.evemisslab.com/en/questions/page/53/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations/#q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0043) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0019-10418 · q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0043 ## Original question and givens 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?). - G001 — 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?). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-sigma-algebra-measure-space](https://ocme.evemisslab.com/en/mko/mko-sigma-algebra-measure-space/) ## Step-by-step derivation S001 · impliesBefore carrier Fin(7);raw family=[[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]] After already canonical family=[[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]] Conditions: none Each set must already be sorted/distinct,and the family mask-ordered and duplicate-free.This protocol rejects noncanonical data rather than reordering aligned measure values to repair it. Step identifiers and dependenciesRule: finite_sigma_canonical_alignment_guard · Uses: G001 S002 · impliesBefore source member=[];fixed carrier=[0, 1, 2, 3, 4, 5, 6] After complement=[0, 1, 2, 3, 4, 5, 6];present=Yes Conditions: none Take the complement in this same carrier for every member and retain every missing-complement record. Step identifiers and dependenciesRule: sigma_every_relative_complement · Uses: G001, S001 S003 · impliesBefore ordered left=[];ordered right=[] After union=[];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S002 S004 · impliesBefore ordered left=[];ordered right=[1, 4] After union=[1, 4];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S003 S005 · impliesBefore ordered left=[];ordered right=[0, 2, 3, 5, 6] After union=[0, 2, 3, 5, 6];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S004 S006 · impliesBefore ordered left=[];ordered right=[0, 1, 2, 3, 4, 5, 6] After union=[0, 1, 2, 3, 4, 5, 6];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S005 S007 · impliesBefore source member=[1, 4];fixed carrier=[0, 1, 2, 3, 4, 5, 6] After complement=[0, 2, 3, 5, 6];present=Yes Conditions: none Take the complement in this same carrier for every member and retain every missing-complement record. Step identifiers and dependenciesRule: sigma_every_relative_complement · Uses: G001, S006 S008 · impliesBefore ordered left=[1, 4];ordered right=[] After union=[1, 4];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S007 S009 · impliesBefore ordered left=[1, 4];ordered right=[1, 4] After union=[1, 4];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S008 S010 · impliesBefore ordered left=[1, 4];ordered right=[0, 2, 3, 5, 6] After union=[0, 1, 2, 3, 4, 5, 6];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S009 S011 · impliesBefore ordered left=[1, 4];ordered right=[0, 1, 2, 3, 4, 5, 6] After union=[0, 1, 2, 3, 4, 5, 6];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S010 S012 · impliesBefore source member=[0, 2, 3, 5, 6];fixed carrier=[0, 1, 2, 3, 4, 5, 6] After complement=[1, 4];present=Yes Conditions: none Take the complement in this same carrier for every member and retain every missing-complement record. Step identifiers and dependenciesRule: sigma_every_relative_complement · Uses: G001, S011 S013 · impliesBefore ordered left=[0, 2, 3, 5, 6];ordered right=[] After union=[0, 2, 3, 5, 6];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S012 S014 · impliesBefore ordered left=[0, 2, 3, 5, 6];ordered right=[1, 4] After union=[0, 1, 2, 3, 4, 5, 6];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S013 S015 · impliesBefore ordered left=[0, 2, 3, 5, 6];ordered right=[0, 2, 3, 5, 6] After union=[0, 2, 3, 5, 6];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S014 S016 · impliesBefore ordered left=[0, 2, 3, 5, 6];ordered right=[0, 1, 2, 3, 4, 5, 6] After union=[0, 1, 2, 3, 4, 5, 6];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S015 S017 · impliesBefore source member=[0, 1, 2, 3, 4, 5, 6];fixed carrier=[0, 1, 2, 3, 4, 5, 6] After complement=[];present=Yes Conditions: none Take the complement in this same carrier for every member and retain every missing-complement record. Step identifiers and dependenciesRule: sigma_every_relative_complement · Uses: G001, S016 S018 · impliesBefore ordered left=[0, 1, 2, 3, 4, 5, 6];ordered right=[] After union=[0, 1, 2, 3, 4, 5, 6];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S017 S019 · impliesBefore ordered left=[0, 1, 2, 3, 4, 5, 6];ordered right=[1, 4] After union=[0, 1, 2, 3, 4, 5, 6];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S018 S020 · impliesBefore ordered left=[0, 1, 2, 3, 4, 5, 6];ordered right=[0, 2, 3, 5, 6] After union=[0, 1, 2, 3, 4, 5, 6];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S019 S021 · impliesBefore ordered left=[0, 1, 2, 3, 4, 5, 6];ordered right=[0, 1, 2, 3, 4, 5, 6] After union=[0, 1, 2, 3, 4, 5, 6];present=Yes Conditions: none Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions. Step identifiers and dependenciesRule: sigma_every_ordered_binary_union · Uses: G001, S020 S022 · impliesBefore empty present=Yes;full present=Yes;missing complements=[];missing unions=[] After finite countable reduction=Yes;sigma qualified=Yes Conditions: none A fixed finite carrier has finitely many distinct subsets,so a countable family's union equals a finite distinct-member union.Empty/whole membership,complements and binary unions therefore suffice here,not for arbitrary infinite carriers. Step identifiers and dependenciesRule: finite_sigma_countable_union_reduction · Uses: G001, S021 ## Reverse checks - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (7, [[], [1, 4], [0, 2, 3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: f9c29d546165f1956c132a7188353944487978a52bcf71e8a025dba0a0136d03 - candidate record SHA-256: a3165f1e65c53c08fc4ce1d4e0b28a65595f849030c93b9cebb722607ae00877 - semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5 - question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0043.json)