# Question 10473: A Finite Measure Need Not Have Total Mass One [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-0098) 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. R0019-10473 · q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0098 ## Original question and givens 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. - G001 — 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. ## 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=[[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]] After already canonical family=[[], [0, 1, 2, 4], [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=[0, 1, 2, 4] After union=[0, 1, 2, 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=[3, 5, 6] After union=[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=[0, 1, 2, 4];fixed carrier=[0, 1, 2, 3, 4, 5, 6] After complement=[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=[0, 1, 2, 4];ordered right=[] After union=[0, 1, 2, 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=[0, 1, 2, 4];ordered right=[0, 1, 2, 4] After union=[0, 1, 2, 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=[0, 1, 2, 4];ordered right=[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=[0, 1, 2, 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=[3, 5, 6];fixed carrier=[0, 1, 2, 3, 4, 5, 6] After complement=[0, 1, 2, 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=[3, 5, 6];ordered right=[] After union=[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=[3, 5, 6];ordered right=[0, 1, 2, 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=[3, 5, 6];ordered right=[3, 5, 6] After union=[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=[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=[0, 1, 2, 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=[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 S023 · impliesBefore canonical row index=0;set=[];raw rational=[0, 1] After aligned normalized measure=0;nonnegative=Yes Conditions: none The original denominator is nonzero and the normalized value must be nonnegative.Bind each value to its original canonical row,not a sorted mass list or another set's mass. Step identifiers and dependenciesRule: measure_original_family_value_pairing · Uses: G001, S022 S024 · impliesBefore canonical row index=1;set=[0, 1, 2, 4];raw rational=[23, 47] After aligned normalized measure=23/47;nonnegative=Yes Conditions: none The original denominator is nonzero and the normalized value must be nonnegative.Bind each value to its original canonical row,not a sorted mass list or another set's mass. Step identifiers and dependenciesRule: measure_original_family_value_pairing · Uses: G001, S023 S025 · impliesBefore canonical row index=2;set=[3, 5, 6];raw rational=[24, 47] After aligned normalized measure=24/47;nonnegative=Yes Conditions: none The original denominator is nonzero and the normalized value must be nonnegative.Bind each value to its original canonical row,not a sorted mass list or another set's mass. Step identifiers and dependenciesRule: measure_original_family_value_pairing · Uses: G001, S024 S026 · impliesBefore canonical row index=3;set=[0, 1, 2, 3, 4, 5, 6];raw rational=[1, 1] After aligned normalized measure=1;nonnegative=Yes Conditions: none The original denominator is nonzero and the normalized value must be nonnegative.Bind each value to its original canonical row,not a sorted mass list or another set's mass. Step identifiers and dependenciesRule: measure_original_family_value_pairing · Uses: G001, S025 S027 · impliesBefore empty-set measure=0 After empty-set zero=Yes Conditions: none Zero empty-set measure is a separate output obligation,not waived by total mass or selected additivity samples. Step identifiers and dependenciesRule: measure_empty_zero_separate_obligation · Uses: G001, S026 S028 · calculationBefore measure sum rows0,0 := sum(0, 0) After measure sum rows0,0 = 0 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore disjoint ordered pair=[[], []];union=[];mu(union)=0;mu(A)=0;mu(B)=0 After sum=0;additivity=Yes Conditions: none Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure. Step identifiers and dependenciesRule: measure_all_disjoint_ordered_pairs · Uses: G001, S028 S030 · calculationBefore measure sum rows0,1 := sum(0, 23/47) After measure sum rows0,1 = 23/47 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · impliesBefore disjoint ordered pair=[[], [0, 1, 2, 4]];union=[0, 1, 2, 4];mu(union)=23/47;mu(A)=0;mu(B)=23/47 After sum=23/47;additivity=Yes Conditions: none Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure. Step identifiers and dependenciesRule: measure_all_disjoint_ordered_pairs · Uses: G001, S030 S032 · calculationBefore measure sum rows0,2 := sum(0, 24/47) After measure sum rows0,2 = 24/47 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · impliesBefore disjoint ordered pair=[[], [3, 5, 6]];union=[3, 5, 6];mu(union)=24/47;mu(A)=0;mu(B)=24/47 After sum=24/47;additivity=Yes Conditions: none Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure. Step identifiers and dependenciesRule: measure_all_disjoint_ordered_pairs · Uses: G001, S032 S034 · calculationBefore measure sum rows0,3 := sum(0, 1) After measure sum rows0,3 = 1 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · impliesBefore disjoint ordered pair=[[], [0, 1, 2, 3, 4, 5, 6]];union=[0, 1, 2, 3, 4, 5, 6];mu(union)=1;mu(A)=0;mu(B)=1 After sum=1;additivity=Yes Conditions: none Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure. Step identifiers and dependenciesRule: measure_all_disjoint_ordered_pairs · Uses: G001, S034 S036 · calculationBefore measure sum rows1,0 := sum(23/47, 0) After measure sum rows1,0 = 23/47 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · impliesBefore disjoint ordered pair=[[0, 1, 2, 4], []];union=[0, 1, 2, 4];mu(union)=23/47;mu(A)=23/47;mu(B)=0 After sum=23/47;additivity=Yes Conditions: none Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure. Step identifiers and dependenciesRule: measure_all_disjoint_ordered_pairs · Uses: G001, S036 S038 · calculationBefore measure sum rows1,2 := sum(23/47, 24/47) After measure sum rows1,2 = 1 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037 S039 · impliesBefore disjoint ordered pair=[[0, 1, 2, 4], [3, 5, 6]];union=[0, 1, 2, 3, 4, 5, 6];mu(union)=1;mu(A)=23/47;mu(B)=24/47 After sum=1;additivity=Yes Conditions: none Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure. Step identifiers and dependenciesRule: measure_all_disjoint_ordered_pairs · Uses: G001, S038 S040 · calculationBefore measure sum rows2,0 := sum(24/47, 0) After measure sum rows2,0 = 24/47 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · impliesBefore disjoint ordered pair=[[3, 5, 6], []];union=[3, 5, 6];mu(union)=24/47;mu(A)=24/47;mu(B)=0 After sum=24/47;additivity=Yes Conditions: none Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure. Step identifiers and dependenciesRule: measure_all_disjoint_ordered_pairs · Uses: G001, S040 S042 · calculationBefore measure sum rows2,1 := sum(24/47, 23/47) After measure sum rows2,1 = 1 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · impliesBefore disjoint ordered pair=[[3, 5, 6], [0, 1, 2, 4]];union=[0, 1, 2, 3, 4, 5, 6];mu(union)=1;mu(A)=24/47;mu(B)=23/47 After sum=1;additivity=Yes Conditions: none Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure. Step identifiers and dependenciesRule: measure_all_disjoint_ordered_pairs · Uses: G001, S042 S044 · calculationBefore measure sum rows3,0 := sum(1, 0) After measure sum rows3,0 = 1 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · impliesBefore disjoint ordered pair=[[0, 1, 2, 3, 4, 5, 6], []];union=[0, 1, 2, 3, 4, 5, 6];mu(union)=1;mu(A)=1;mu(B)=0 After sum=1;additivity=Yes Conditions: none Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure. Step identifiers and dependenciesRule: measure_all_disjoint_ordered_pairs · Uses: G001, S044 S046 · impliesBefore sigma qualified=Yes;empty zero=Yes;all bad disjoint pairs=[] After finite countable-additivity reduction=Yes Conditions: none On a finite carrier there are at most n pairwise disjoint nonempty sets;all remaining countable terms are empty with mass0.Complete finite disjoint additivity therefore handles this carrier's countable case,not arbitrary infinite measure spaces. Step identifiers and dependenciesRule: finite_measure_countable_disjoint_reduction · Uses: G001, S045 S047 · impliesBefore finite measure qualified=Yes;total carrier measure=1 After probability measure=Yes Conditions: none A probability measure additionally needs total mass exactly1.A valid measure of total2 is not a probability measure,and total1 does not repair failed additivity or nonzero empty mass. Step identifiers and dependenciesRule: probability_normalization_after_measure_obligations · Uses: G001, S046 ## Reverse checks - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S044 · premise_and_result_audit — Recompute measure sum rows3,0 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Recompute measure sum rows2,1 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S040 · premise_and_result_audit — Recompute measure sum rows2,0 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Recompute measure sum rows1,2 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Recompute measure sum rows1,0 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S034 · premise_and_result_audit — Recompute measure sum rows0,3 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Recompute measure sum rows0,2 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Recompute measure sum rows0,1 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Recompute measure sum rows0,0 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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, [[], [0, 1, 2, 4], [3, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 23/47, 24/47, 1], Yes, [], 1, Yes, 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: 8ce70b99f6733995f9157c5e6404016ecd9251abcfd047e2cfe665f8b8936fc9 - candidate record SHA-256: c5b4fbf742900c547c14775b1f8d5dab39223bfff8fbf33be0344b9ebbdd4bcd - 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-0098.json)