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R0019-10426 · q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0051
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.
G001 — 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.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.
Before
carrier Fin(7);raw family=[[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]
After
already canonical family=[[], [0], [1, 2, 3, 4, 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.
Rule: finite_sigma_canonical_alignment_guard · Uses: G001
Before
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.
Rule: sigma_every_relative_complement · Uses: G001, S001
Before
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.
Rule: sigma_every_ordered_binary_union · Uses: G001, S002
Before
ordered left=[];ordered right=[0]
After
union=[0];present=Yes
Conditions: none
Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.
Rule: sigma_every_ordered_binary_union · Uses: G001, S003
Before
ordered left=[];ordered right=[1, 2, 3, 4, 5, 6]
After
union=[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.
Rule: sigma_every_ordered_binary_union · Uses: G001, S004
Before
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.
Rule: sigma_every_ordered_binary_union · Uses: G001, S005
Before
source member=[0];fixed carrier=[0, 1, 2, 3, 4, 5, 6]
After
complement=[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.
Rule: sigma_every_relative_complement · Uses: G001, S006
Before
ordered left=[0];ordered right=[]
After
union=[0];present=Yes
Conditions: none
Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.
Rule: sigma_every_ordered_binary_union · Uses: G001, S007
Before
ordered left=[0];ordered right=[0]
After
union=[0];present=Yes
Conditions: none
Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.
Rule: sigma_every_ordered_binary_union · Uses: G001, S008
Before
ordered left=[0];ordered right=[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.
Rule: sigma_every_ordered_binary_union · Uses: G001, S009
Before
ordered left=[0];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.
Rule: sigma_every_ordered_binary_union · Uses: G001, S010
Before
source member=[1, 2, 3, 4, 5, 6];fixed carrier=[0, 1, 2, 3, 4, 5, 6]
After
complement=[0];present=Yes
Conditions: none
Take the complement in this same carrier for every member and retain every missing-complement record.
Rule: sigma_every_relative_complement · Uses: G001, S011
Before
ordered left=[1, 2, 3, 4, 5, 6];ordered right=[]
After
union=[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.
Rule: sigma_every_ordered_binary_union · Uses: G001, S012
Before
ordered left=[1, 2, 3, 4, 5, 6];ordered right=[0]
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.
Rule: sigma_every_ordered_binary_union · Uses: G001, S013
Before
ordered left=[1, 2, 3, 4, 5, 6];ordered right=[1, 2, 3, 4, 5, 6]
After
union=[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.
Rule: sigma_every_ordered_binary_union · Uses: G001, S014
Before
ordered left=[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.
Rule: sigma_every_ordered_binary_union · Uses: G001, S015
Before
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.
Rule: sigma_every_relative_complement · Uses: G001, S016
Before
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.
Rule: sigma_every_ordered_binary_union · Uses: G001, S017
Before
ordered left=[0, 1, 2, 3, 4, 5, 6];ordered right=[0]
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.
Rule: sigma_every_ordered_binary_union · Uses: G001, S018
Before
ordered left=[0, 1, 2, 3, 4, 5, 6];ordered right=[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.
Rule: sigma_every_ordered_binary_union · Uses: G001, S019
Before
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.
Rule: sigma_every_ordered_binary_union · Uses: G001, S020
Before
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.
Rule: finite_sigma_countable_union_reduction · Uses: G001, S021
Before
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.
Rule: measure_original_family_value_pairing · Uses: G001, S022
Before
canonical row index=1;set=[0];raw rational=[1, 3]
After
aligned normalized measure=1/3;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.
Rule: measure_original_family_value_pairing · Uses: G001, S023
Before
canonical row index=2;set=[1, 2, 3, 4, 5, 6];raw rational=[2, 3]
After
aligned normalized measure=2/3;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.
Rule: measure_original_family_value_pairing · Uses: G001, S024
Before
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.
Rule: measure_original_family_value_pairing · Uses: G001, S025
Before
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.
Rule: measure_empty_zero_separate_obligation · Uses: G001, S026
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
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.
Rule: measure_all_disjoint_ordered_pairs · Uses: G001, S028
Before
measure sum rows0,1 := sum(0, 1/3)
After
measure sum rows0,1 = 1/3
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
disjoint ordered pair=[[], [0]];union=[0];mu(union)=1/3;mu(A)=0;mu(B)=1/3
After
sum=1/3;additivity=Yes
Conditions: none
Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure.
Rule: measure_all_disjoint_ordered_pairs · Uses: G001, S030
Before
measure sum rows0,2 := sum(0, 2/3)
After
measure sum rows0,2 = 2/3
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
disjoint ordered pair=[[], [1, 2, 3, 4, 5, 6]];union=[1, 2, 3, 4, 5, 6];mu(union)=2/3;mu(A)=0;mu(B)=2/3
After
sum=2/3;additivity=Yes
Conditions: none
Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure.
Rule: measure_all_disjoint_ordered_pairs · Uses: G001, S032
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
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.
Rule: measure_all_disjoint_ordered_pairs · Uses: G001, S034
Before
measure sum rows1,0 := sum(1/3, 0)
After
measure sum rows1,0 = 1/3
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
disjoint ordered pair=[[0], []];union=[0];mu(union)=1/3;mu(A)=1/3;mu(B)=0
After
sum=1/3;additivity=Yes
Conditions: none
Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure.
Rule: measure_all_disjoint_ordered_pairs · Uses: G001, S036
Before
measure sum rows1,2 := sum(1/3, 2/3)
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.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
disjoint ordered pair=[[0], [1, 2, 3, 4, 5, 6]];union=[0, 1, 2, 3, 4, 5, 6];mu(union)=1;mu(A)=1/3;mu(B)=2/3
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.
Rule: measure_all_disjoint_ordered_pairs · Uses: G001, S038
Before
measure sum rows2,0 := sum(2/3, 0)
After
measure sum rows2,0 = 2/3
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
disjoint ordered pair=[[1, 2, 3, 4, 5, 6], []];union=[1, 2, 3, 4, 5, 6];mu(union)=2/3;mu(A)=2/3;mu(B)=0
After
sum=2/3;additivity=Yes
Conditions: none
Include ordered pairs involving the empty set.Apply binary additivity to disjoint sets only and retain every failure.
Rule: measure_all_disjoint_ordered_pairs · Uses: G001, S040
Before
measure sum rows2,1 := sum(2/3, 1/3)
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.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
disjoint ordered pair=[[1, 2, 3, 4, 5, 6], [0]];union=[0, 1, 2, 3, 4, 5, 6];mu(union)=1;mu(A)=2/3;mu(B)=1/3
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.
Rule: measure_all_disjoint_ordered_pairs · Uses: G001, S042
Before
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.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
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.
Rule: measure_all_disjoint_ordered_pairs · Uses: G001, S044
Before
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.
Rule: finite_measure_countable_disjoint_reduction · Uses: G001, S045
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.(7, [[], [0], [1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [0, 1/3, 2/3, 1], Yes, [], Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: cd50017ec0ca60f7f3ccbcdd0ea441da3b982a6104a6ca51148be3205d899e21candidate record SHA-256: 7e0709a586744dc0c19f2b7517624c18da2b59ee9009df6c107d7887a1fc93d0semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419Lean: NOT_RUN · human review: NOT_PERFORMED