# Complete Finite Measurability through Target Preimages

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261013-advance-20261024-r02-measurable-function-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. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[0, 0, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [0, 0, 0, 0], [0, 1], [0, 1, 2, 3])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 0, 0, 0], [0, 1], [0, 1, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0001/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 2. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[1, 0, 0, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [1, 0, 0, 0], [1, 2], [0])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 0, 0, 0], [1, 2], [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0002/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 3. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[2, 0, 0, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [2, 0, 0, 0], [2, 3], [0])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 0, 0, 0], [2, 3], [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0003/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 4. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[3, 0, 0, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [3, 0, 0, 0], [0, 3], [0, 1, 2, 3])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 0, 0, 0], [0, 3], [0, 1, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0004/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 5. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[0, 1, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [0, 1, 0, 0], [0, 1], [0, 1, 2, 3])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 1, 0, 0], [0, 1], [0, 1, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0005/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 6. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[1, 1, 0, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [1, 1, 0, 0], [1, 2], [0, 1])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 1, 0, 0], [1, 2], [0, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0006/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 7. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[2, 1, 0, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [2, 1, 0, 0], [2, 3], [0])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 1, 0, 0], [2, 3], [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0007/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 8. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[3, 1, 0, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [3, 1, 0, 0], [0, 3], [0, 2, 3])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 1, 0, 0], [0, 3], [0, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0008/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 9. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[0, 2, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [0, 2, 0, 0], [0, 1], [0, 2, 3])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 2, 0, 0], [0, 1], [0, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0009/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 10. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[1, 2, 0, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [1, 2, 0, 0], [1, 2], [0, 1])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 2, 0, 0], [1, 2], [0, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0010/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 11. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[2, 2, 0, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [2, 2, 0, 0], [2, 3], [0, 1])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 2, 0, 0], [2, 3], [0, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0011/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 12. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[3, 2, 0, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [3, 2, 0, 0], [0, 3], [0, 2, 3])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 2, 0, 0], [0, 3], [0, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0012/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 13. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[0, 3, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [0, 3, 0, 0], [0, 1], [0, 2, 3])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 3, 0, 0], [0, 1], [0, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0013/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 14. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[1, 3, 0, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [1, 3, 0, 0], [1, 2], [0])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 3, 0, 0], [1, 2], [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0014/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 15. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[2, 3, 0, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [2, 3, 0, 0], [2, 3], [0, 1])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 3, 0, 0], [2, 3], [0, 1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0015/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 16. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[3, 3, 0, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [3, 3, 0, 0], [0, 3], [0, 1, 2, 3])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 3, 0, 0], [0, 3], [0, 1, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0016/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 17. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[0, 0, 1, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [0, 0, 1, 0], [0, 1], [0, 1, 2, 3])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 0, 1, 0], [0, 1], [0, 1, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0017/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 18. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[1, 0, 1, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [1, 0, 1, 0], [1, 2], [0, 2])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 0, 1, 0], [1, 2], [0, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0018/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 19. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[2, 0, 1, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [2, 0, 1, 0], [2, 3], [0])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 0, 1, 0], [2, 3], [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0019/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 20. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[3, 0, 1, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [3, 0, 1, 0], [0, 3], [0, 1, 3])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 0, 1, 0], [0, 3], [0, 1, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0020/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 21. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[0, 1, 1, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [0, 1, 1, 0], [0, 1], [0, 1, 2, 3])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 1, 1, 0], [0, 1], [0, 1, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0021/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 22. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[1, 1, 1, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [1, 1, 1, 0], [1, 2], [0, 1, 2])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 1, 1, 0], [1, 2], [0, 1, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0022/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 23. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[2, 1, 1, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [2, 1, 1, 0], [2, 3], [0])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 1, 1, 0], [2, 3], [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0023/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 24. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[3, 1, 1, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [3, 1, 1, 0], [0, 3], [0, 3])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 1, 1, 0], [0, 3], [0, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0024/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 25. The Inverse Image under a Complete Finite Total Map

Use carriers Fin(4) and Fin(4), complete total mapping=[0, 2, 1, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

View answer and explanation(4, 4, [0, 2, 1, 0], [0, 1], [0, 2, 3])

Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 2, 1, 0], [0, 1], [0, 2, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0025/)

Measurable function · qf-mod24-measurable-preimage · candidate





## 26. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 0, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0026/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 27. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 0, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [1, 0, 0, 0], [[[], [], Yes], [[0], [1, 2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 0, 0, 0], [[[], [], Yes], [[0], [1, 2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0027/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 28. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 0, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [2, 0, 0, 0], [[[], [], Yes], [[0], [1, 2, 3], No], [[1], [], Yes], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], No], [[1, 3], [], Yes], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 0, 0, 0], [[[], [], Yes], [[0], [1, 2, 3], No], [[1], [], Yes], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], No], [[1, 3], [], Yes], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0028/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 29. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 0, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [3, 0, 0, 0], [[[], [], Yes], [[0], [1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 0, 0, 0], [[[], [], Yes], [[0], [1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0029/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 30. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 1, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [0, 1, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], No], [[1], [1], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], No], [[1, 3], [1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 1, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], No], [[1], [1], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], No], [[1, 3], [1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0030/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 31. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 1, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [1, 1, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0, 1], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 1, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0, 1], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0031/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 32. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 1, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [2, 1, 0, 0], [[[], [], Yes], [[0], [2, 3], No], [[1], [1], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 2, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], No], [[1, 3], [1], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 1, 0, 0], [[[], [], Yes], [[0], [2, 3], No], [[1], [1], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 2, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], No], [[1, 3], [1], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0032/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 33. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 1, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [3, 1, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [1], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 1, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [1], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0033/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 34. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 2, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [0, 2, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 2, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0034/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 35. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 2, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [1, 2, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 2, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0035/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 36. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 2, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [2, 2, 0, 0], [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [0, 1], No], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], No], [[1, 3], [], Yes], [[0, 1, 3], [2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 2, 0, 0], [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [0, 1], No], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], No], [[1, 3], [], Yes], [[0, 1, 3], [2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0036/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 37. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 2, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [3, 2, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [1], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 2, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [1], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0037/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 38. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [1, 2], [0, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 3, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [0, 3, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], No], [[1], [], Yes], [[0, 1], [0, 2, 3], No], [[2], [], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 3, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], No], [[1], [], Yes], [[0, 1], [0, 2, 3], No], [[2], [], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0038/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 39. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 3, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [1, 3, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 3, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0039/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 40. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 3, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [2, 3, 0, 0], [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [0], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [0], Yes], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 2, 3], Yes], [[1, 3], [1], No], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 3, 0, 0], [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [0], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [0], Yes], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 2, 3], Yes], [[1, 3], [1], No], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0040/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 41. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 3, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [3, 3, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [2, 3], Yes], [[3], [0, 1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 3, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [2, 3], Yes], [[3], [0, 1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0041/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 42. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 0, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [0, 0, 1, 0], [[[], [], Yes], [[0], [0, 1, 3], No], [[1], [2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 3], No], [[1, 2], [2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 3], No], [[1, 3], [2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 0, 1, 0], [[[], [], Yes], [[0], [0, 1, 3], No], [[1], [2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 3], No], [[1, 2], [2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 3], No], [[1, 3], [2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0042/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 43. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 0, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [1, 0, 1, 0], [[[], [], Yes], [[0], [1, 3], Yes], [[1], [0, 2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 3], Yes], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 0, 1, 0], [[[], [], Yes], [[0], [1, 3], Yes], [[1], [0, 2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 3], Yes], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0043/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 44. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 0, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [2, 0, 1, 0], [[[], [], Yes], [[0], [1, 3], Yes], [[1], [2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 1, 3], No], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], Yes], [[1, 3], [2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 0, 1, 0], [[[], [], Yes], [[0], [1, 3], Yes], [[1], [2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 1, 3], No], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], Yes], [[1, 3], [2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0044/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 45. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 0, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [3, 0, 1, 0], [[[], [], Yes], [[0], [1, 3], Yes], [[1], [2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [2], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 1, 3], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 0, 1, 0], [[[], [], Yes], [[0], [1, 3], Yes], [[1], [2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [2], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 1, 3], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0045/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 46. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 1, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [0, 1, 1, 0], [[[], [], Yes], [[0], [0, 3], No], [[1], [1, 2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 1, 1, 0], [[[], [], Yes], [[0], [0, 3], No], [[1], [1, 2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0046/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 47. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 1, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [1, 1, 1, 0], [[[], [], Yes], [[0], [3], Yes], [[1], [0, 1, 2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [3], Yes], [[1, 2], [0, 1, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], Yes], [[1, 3], [0, 1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [3], Yes], [[1, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 1, 1, 0], [[[], [], Yes], [[0], [3], Yes], [[1], [0, 1, 2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [3], Yes], [[1, 2], [0, 1, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], Yes], [[1, 3], [0, 1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [3], Yes], [[1, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0047/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 48. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 1, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [2, 1, 1, 0], [[[], [], Yes], [[0], [3], No], [[1], [1, 2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 1, 1, 0], [[[], [], Yes], [[0], [3], No], [[1], [1, 2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0048/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 49. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 1, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [3, 1, 1, 0], [[[], [], Yes], [[0], [3], Yes], [[1], [1, 2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [3], Yes], [[1, 2], [1, 2], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 3], Yes], [[1, 3], [0, 1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 3], Yes], [[1, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 1, 1, 0], [[[], [], Yes], [[0], [3], Yes], [[1], [1, 2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [3], Yes], [[1, 2], [1, 2], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 3], Yes], [[1, 3], [0, 1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 3], Yes], [[1, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0049/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 50. Audit Every Target Measurable Set by Preimage

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 2, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

View answer and explanation(4, 4, [0, 2, 1, 0], [[[], [], Yes], [[0], [0, 3], No], [[1], [2], Yes], [[0, 1], [0, 2, 3], No], [[2], [1], No], [[0, 2], [0, 1, 3], Yes], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], No], [[1, 3], [2], Yes], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1], No], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 2, 1, 0], [[[], [], Yes], [[0], [0, 3], No], [[1], [2], Yes], [[0, 1], [0, 2, 3], No], [[2], [1], No], [[0, 2], [0, 1, 3], Yes], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], No], [[1, 3], [2], Yes], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1], No], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0050/)

Measurable function · qf-mod24-measurable-map-audit · candidate





## 51. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0], [1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0], [1, 0, 0, 0, 0, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0051/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 52. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1], [0, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[1] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [1], [0, 1, 0, 0, 0, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [1], [0, 1, 0, 0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0052/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 53. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1], [2, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0, 1] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0, 1], [1, 1, 0, 0, 0, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 1], [1, 1, 0, 0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0053/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 54. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [2], [0, 1, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0], [1, 0, 0, 0, 0, 0], No)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0054/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 55. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 2], [1, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0, 2] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0, 2], [1, 0, 1, 0, 0, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 2], [1, 0, 1, 0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0055/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 56. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1, 2], [0, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[1, 2] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [1, 2], [0, 1, 1, 0, 0, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [1, 2], [0, 1, 1, 0, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0056/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 57. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 2], [3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0], [1, 0, 0, 0, 0, 0], No)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0057/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 58. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [3], [0, 1, 2, 4, 5], [0, 1, 2, 3, 4, 5]], event=[3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [3], [0, 0, 0, 1, 0, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [3], [0, 0, 0, 1, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0058/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 59. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 3], [1, 2, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0, 3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0, 3], [1, 0, 0, 1, 0, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 3], [1, 0, 0, 1, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0059/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 60. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1, 3], [0, 2, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0], [1, 0, 0, 0, 0, 0], No)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0060/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 61. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 3], [2, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0, 1, 3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0, 1, 3], [1, 1, 0, 1, 0, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 1, 3], [1, 1, 0, 1, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0061/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 62. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [2, 3], [0, 1, 4, 5], [0, 1, 2, 3, 4, 5]], event=[2, 3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [2, 3], [0, 0, 1, 1, 0, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [2, 3], [0, 0, 1, 1, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0062/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 63. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 2, 3], [1, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0], [1, 0, 0, 0, 0, 0], No)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0063/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 64. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1, 2, 3], [0, 4, 5], [0, 1, 2, 3, 4, 5]], event=[1, 2, 3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [1, 2, 3], [0, 1, 1, 1, 0, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [1, 2, 3], [0, 1, 1, 1, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0064/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 65. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 2, 3], [4, 5], [0, 1, 2, 3, 4, 5]], event=[0, 1, 2, 3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0, 1, 2, 3], [1, 1, 1, 1, 0, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 1, 2, 3], [1, 1, 1, 1, 0, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0065/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 66. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [4], [0, 1, 2, 3, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0], [1, 0, 0, 0, 0, 0], No)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0066/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 67. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 4], [1, 2, 3, 5], [0, 1, 2, 3, 4, 5]], event=[0, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0, 4], [1, 0, 0, 0, 1, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 4], [1, 0, 0, 0, 1, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0067/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 68. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1, 4], [0, 2, 3, 5], [0, 1, 2, 3, 4, 5]], event=[1, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [1, 4], [0, 1, 0, 0, 1, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [1, 4], [0, 1, 0, 0, 1, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0068/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 69. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 4], [2, 3, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0], [1, 0, 0, 0, 0, 0], No)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0069/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 70. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [2, 4], [0, 1, 3, 5], [0, 1, 2, 3, 4, 5]], event=[2, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [2, 4], [0, 0, 1, 0, 1, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [2, 4], [0, 0, 1, 0, 1, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0070/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 71. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 2, 4], [1, 3, 5], [0, 1, 2, 3, 4, 5]], event=[0, 2, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0, 2, 4], [1, 0, 1, 0, 1, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 2, 4], [1, 0, 1, 0, 1, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0071/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 72. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1, 2, 4], [0, 3, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0], [1, 0, 0, 0, 0, 0], No)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0072/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 73. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 2, 4], [3, 5], [0, 1, 2, 3, 4, 5]], event=[0, 1, 2, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0, 1, 2, 4], [1, 1, 1, 0, 1, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 1, 2, 4], [1, 1, 1, 0, 1, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0073/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 74. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [3, 4], [0, 1, 2, 5], [0, 1, 2, 3, 4, 5]], event=[3, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [3, 4], [0, 0, 0, 1, 1, 0], Yes)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [3, 4], [0, 0, 0, 1, 1, 0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0074/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 75. An Indicator’s Measurability Depends on Its Event

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 3, 4], [1, 2, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).

View answer and explanation(6, [0], [1, 0, 0, 0, 0, 0], No)

The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0075/)

Measurable function · qf-mod24-measurable-indicator-audit · candidate





## 76. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[0, 0, 0, 0], second=[1, 1, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(Yes, Yes, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0076/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 77. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[1, 0, 0, 0], second=[2, 1, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(Yes, Yes, [1, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [0], Yes], [[0, 1], [0], Yes], [[2], [1, 2, 3], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0], Yes], [[2, 3], [1, 2, 3], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [1, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [0], Yes], [[0, 1], [0], Yes], [[2], [1, 2, 3], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0], Yes], [[2, 3], [1, 2, 3], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0077/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 78. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[2, 0, 0, 0], second=[3, 1, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(No, Yes, [2, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0], No], [[0, 2], [0], No], [[1, 2], [0], No], [[0, 1, 2], [0], No], [[3], [1, 2, 3], No], [[0, 3], [1, 2, 3], No], [[1, 3], [1, 2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [2, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0], No], [[0, 2], [0], No], [[1, 2], [0], No], [[0, 1, 2], [0], No], [[3], [1, 2, 3], No], [[0, 3], [1, 2, 3], No], [[1, 3], [1, 2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0078/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 79. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[3, 0, 0, 0], second=[0, 2, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(Yes, Yes, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0079/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 80. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[0, 1, 0, 0], second=[1, 2, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(No, Yes, [1, 2, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2, 3], No], [[0, 1], [0, 2, 3], No], [[2], [1], No], [[0, 2], [1], No], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1], No], [[0, 2, 3], [1], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [1, 2, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2, 3], No], [[0, 1], [0, 2, 3], No], [[2], [1], No], [[0, 2], [1], No], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1], No], [[0, 2, 3], [1], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0080/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 81. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[1, 1, 0, 0], second=[2, 2, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(Yes, Yes, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0081/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 82. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[2, 1, 0, 0], second=[3, 2, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(No, Yes, [2, 2, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1], No], [[0, 2], [0, 1], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1], No], [[3], [2, 3], No], [[0, 3], [2, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [2, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [2, 2, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1], No], [[0, 2], [0, 1], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1], No], [[3], [2, 3], No], [[0, 3], [2, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [2, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0082/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 83. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[3, 1, 0, 0], second=[0, 3, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(Yes, Yes, [0, 3, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 3, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0083/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 84. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[0, 2, 0, 0], second=[1, 3, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(Yes, Yes, [1, 2, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2, 3], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [1], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2, 3], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [1, 2, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2, 3], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [1], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2, 3], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0084/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 85. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[1, 2, 0, 0], second=[2, 3, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(Yes, Yes, [3, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [1, 2, 3], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [1, 2, 3], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [3, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [1, 2, 3], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [1, 2, 3], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0085/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 86. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[2, 2, 0, 0], second=[3, 3, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(No, Yes, [2, 2, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1], No], [[0, 2], [0, 1], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1], No], [[3], [2, 3], No], [[0, 3], [2, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [2, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [2, 2, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1], No], [[0, 2], [0, 1], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1], No], [[3], [2, 3], No], [[0, 3], [2, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [2, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0086/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 87. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[3, 2, 0, 0], second=[0, 0, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(Yes, Yes, [0, 3, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 3, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0087/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 88. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [1, 2], [0, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[0, 3, 0, 0], second=[1, 0, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(No, Yes, [1, 0, 1, 1], [[[], [], Yes], [[0], [1], No], [[1], [0, 2, 3], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1], No], [[1, 2], [0, 2, 3], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1], No], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [1, 0, 1, 1], [[[], [], Yes], [[0], [1], No], [[1], [0, 2, 3], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1], No], [[1, 2], [0, 2, 3], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1], No], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0088/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 89. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[1, 3, 0, 0], second=[2, 0, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(Yes, Yes, [0, 0, 2, 2], [[[], [], Yes], [[0], [0, 1], Yes], [[1], [], Yes], [[0, 1], [0, 1], Yes], [[2], [2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1], Yes], [[2, 3], [2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 0, 2, 2], [[[], [], Yes], [[0], [0, 1], Yes], [[1], [], Yes], [[0, 1], [0, 1], Yes], [[2], [2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1], Yes], [[2, 3], [2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0089/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 90. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[2, 3, 0, 0], second=[3, 0, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(No, Yes, [3, 0, 3, 3], [[[], [], Yes], [[0], [1], No], [[1], [], Yes], [[0, 1], [1], No], [[2], [], Yes], [[0, 2], [1], No], [[1, 2], [], Yes], [[0, 1, 2], [1], No], [[3], [0, 2, 3], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 2, 3], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [3, 0, 3, 3], [[[], [], Yes], [[0], [1], No], [[1], [], Yes], [[0, 1], [1], No], [[2], [], Yes], [[0, 2], [1], No], [[1, 2], [], Yes], [[0, 1, 2], [1], No], [[3], [0, 2, 3], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 2, 3], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0090/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 91. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[3, 3, 0, 0], second=[0, 1, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(Yes, Yes, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0091/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 92. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[0, 0, 1, 0], second=[1, 1, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(No, Yes, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, No)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0092/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 93. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[1, 0, 1, 0], second=[2, 1, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(Yes, Yes, [1, 2, 1, 2], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2], Yes], [[0, 1], [0, 2], Yes], [[2], [1, 3], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 2], Yes], [[2, 3], [1, 3], Yes], [[0, 2, 3], [1, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [1, 2, 1, 2], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2], Yes], [[0, 1], [0, 2], Yes], [[2], [1, 3], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 2], Yes], [[2, 3], [1, 3], Yes], [[0, 2, 3], [1, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0093/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 94. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[2, 0, 1, 0], second=[3, 1, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(No, Yes, [3, 3, 1, 3], [[[], [], Yes], [[0], [], Yes], [[1], [2], No], [[0, 1], [2], No], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [2], No], [[0, 1, 2], [2], No], [[3], [0, 1, 3], No], [[0, 3], [0, 1, 3], No], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [3, 3, 1, 3], [[[], [], Yes], [[0], [], Yes], [[1], [2], No], [[0, 1], [2], No], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [2], No], [[0, 1, 2], [2], No], [[3], [0, 1, 3], No], [[0, 3], [0, 1, 3], No], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0094/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 95. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[3, 0, 1, 0], second=[0, 2, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(Yes, Yes, [0, 0, 2, 0], [[[], [], Yes], [[0], [0, 1, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 3], Yes], [[2], [2], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 3], Yes], [[2, 3], [2], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 0, 2, 0], [[[], [], Yes], [[0], [0, 1, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 3], Yes], [[2], [2], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 3], Yes], [[2, 3], [2], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0095/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 96. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[0, 1, 1, 0], second=[1, 2, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(No, Yes, [1, 2, 2, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 3], No], [[0, 1], [0, 3], No], [[2], [1, 2], No], [[0, 2], [1, 2], No], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 3], No], [[2, 3], [1, 2], No], [[0, 2, 3], [1, 2], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [1, 2, 2, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 3], No], [[0, 1], [0, 3], No], [[2], [1, 2], No], [[0, 2], [1, 2], No], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 3], No], [[2, 3], [1, 2], No], [[0, 2, 3], [1, 2], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0096/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 97. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[1, 1, 1, 0], second=[2, 2, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(Yes, Yes, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0097/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 98. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[2, 1, 1, 0], second=[3, 2, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(No, Yes, [3, 2, 2, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [1, 2], No], [[0, 2], [1, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [1, 2], No], [[3], [0, 3], No], [[0, 3], [0, 3], No], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [3, 2, 2, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [1, 2], No], [[0, 2], [1, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [1, 2], No], [[3], [0, 3], No], [[0, 3], [0, 3], No], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0098/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 99. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[3, 1, 1, 0], second=[0, 3, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(Yes, Yes, [0, 3, 3, 0], [[[], [], Yes], [[0], [0, 3], Yes], [[1], [], Yes], [[0, 1], [0, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 3], Yes], [[3], [1, 2], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1, 2], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 3, 3, 0], [[[], [], Yes], [[0], [0, 3], Yes], [[1], [], Yes], [[0, 1], [0, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 3], Yes], [[3], [1, 2], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1, 2], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0099/)

Measurable function · qf-mod24-measurable-composition-audit · candidate





## 100. Composition: Individual Checks, Composite Check and Their Conjunction

Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[0, 2, 1, 0], second=[1, 3, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.

View answer and explanation(No, Yes, [1, 3, 3, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 3], No], [[0, 1], [0, 3], No], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 3], No], [[3], [1, 2], No], [[0, 3], [1, 2], No], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1, 2], No], [[0, 2, 3], [1, 2], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)

Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [1, 3, 3, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 3], No], [[0, 1], [0, 3], No], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 3], No], [[3], [1, 2], No], [[0, 3], [1, 2], No], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1, 2], No], [[0, 2, 3], [1, 2], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261024-r02-measurable-function-0100/)

Measurable function · qf-mod24-measurable-composition-audit · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261024-r02-measurable-function-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261024-r02-measurable-function-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261024-r02-measurable-function-foundations/validation-report.json)