# Finite Sigma-Algebra Closure and Exact Measure Reduction

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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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0100/)

Sigma-algebras and measure spaces · qf-mod23-measure-probability-measure-audit · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261023-r05-sigma-algebra-measure-space-foundations/validation-report.json)