# Finite Preimage Laws, Constant Maps and Source-Sigma Refinement

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-20261025-r05-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. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 0, 0, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 0, 0], [0, 1], [2, 3], [0, 1, 2, 3], [], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0001/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 2. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 0, 0, 0], target event=[1, 2] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 0, 0], [1, 2], [0, 3], [0], [1, 2, 3], [1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0002/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 3. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 0, 0, 0], target event=[2, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 0, 0, 0], [2, 3], [0, 1], [0], [1, 2, 3], [1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0003/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 4. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 0, 0, 0], target event=[0, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0, 0, 0], [0, 3], [1, 2], [0, 1, 2, 3], [], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0004/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 5. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 1, 0, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0, 0], [0, 1], [2, 3], [0, 1, 2, 3], [], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0005/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 6. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 1, 0, 0], target event=[1, 2] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 0, 0], [1, 2], [0, 3], [0, 1], [2, 3], [2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0006/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 7. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 1, 0, 0], target event=[2, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 1, 0, 0], [2, 3], [0, 1], [0], [1, 2, 3], [1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0007/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 8. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 1, 0, 0], target event=[0, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 1, 0, 0], [0, 3], [1, 2], [0, 2, 3], [1], [1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0008/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 9. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 2, 0, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2, 0, 0], [0, 1], [2, 3], [0, 2, 3], [1], [1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0009/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 10. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 2, 0, 0], target event=[1, 2] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2, 0, 0], [1, 2], [0, 3], [0, 1], [2, 3], [2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0010/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 11. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 2, 0, 0], target event=[2, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, 0, 0], [2, 3], [0, 1], [0, 1], [2, 3], [2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0011/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 12. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 2, 0, 0], target event=[0, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 0, 0], [0, 3], [1, 2], [0, 2, 3], [1], [1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0012/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 13. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 3, 0, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 3, 0, 0], [0, 1], [2, 3], [0, 2, 3], [1], [1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0013/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 14. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 3, 0, 0], target event=[1, 2] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 3, 0, 0], [1, 2], [0, 3], [0], [1, 2, 3], [1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0014/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 15. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 3, 0, 0], target event=[2, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3, 0, 0], [2, 3], [0, 1], [0, 1], [2, 3], [2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0015/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 16. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 3, 0, 0], target event=[0, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 3, 0, 0], [0, 3], [1, 2], [0, 1, 2, 3], [], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0016/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 17. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 0, 1, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 1, 0], [0, 1], [2, 3], [0, 1, 2, 3], [], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0017/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 18. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 0, 1, 0], target event=[1, 2] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1, 0], [1, 2], [0, 3], [0, 2], [1, 3], [1, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0018/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 19. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 0, 1, 0], target event=[2, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 0, 1, 0], [2, 3], [0, 1], [0], [1, 2, 3], [1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0019/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 20. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 0, 1, 0], target event=[0, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0, 1, 0], [0, 3], [1, 2], [0, 1, 3], [2], [2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0020/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 21. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 1, 1, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 1, 0], [0, 1], [2, 3], [0, 1, 2, 3], [], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0021/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 22. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 1, 1, 0], target event=[1, 2] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 1, 0], [1, 2], [0, 3], [0, 1, 2], [3], [3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0022/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 23. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 1, 1, 0], target event=[2, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 1, 1, 0], [2, 3], [0, 1], [0], [1, 2, 3], [1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0023/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 24. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 1, 1, 0], target event=[0, 3] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 1, 1, 0], [0, 3], [1, 2], [0, 3], [1, 2], [1, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0024/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 25. Complement under a Complete Total-Map Preimage

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 2, 1, 0], target event=[0, 1] normalized as a set. Give (mapping,event,target complement,preimage,source complement of preimage,preimage of target complement,equal?). This family audits a set identity without separately supplied sigma-algebras.

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

Take each complement in its own carrier. Do not confuse the source complement with the target complement. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2, 1, 0], [0, 1], [2, 3], [0, 2, 3], [1], [1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0025/)

Measurable function · qf-mod25-measdeep-preimage-complement-law · candidate





## 26. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 0, 0, 0], target events left=[0], right=[1, 2], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 0, 0], [0], [1, 2], [0, 1, 2], [0, 1, 2, 3], [], [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0026/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 27. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 0, 0, 0], target events left=[1], right=[2, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 0, 0], [1], [2, 3], [1, 2, 3], [0], [], [0], [0], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0027/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 28. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 0, 0, 0], target events left=[2], right=[0, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 0, 0, 0], [2], [0, 3], [0, 2, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0028/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 29. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 0, 0, 0], target events left=[3], right=[0, 1], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0, 0, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0029/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 30. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 1, 0, 0], target events left=[0], right=[1, 2], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [1], [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0030/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 31. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 1, 0, 0], target events left=[1], right=[2, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 0, 0], [1], [2, 3], [1, 2, 3], [0, 1], [], [0, 1], [0, 1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0031/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 32. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 1, 0, 0], target events left=[2], right=[0, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 1, 0, 0], [2], [0, 3], [0, 2, 3], [0], [2, 3], [0, 2, 3], [0, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0032/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 33. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 1, 0, 0], target events left=[3], right=[0, 1], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 1, 0, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0033/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 34. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 2, 0, 0], target events left=[0], right=[1, 2], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [1], [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0034/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 35. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 2, 0, 0], target events left=[1], right=[2, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2, 0, 0], [1], [2, 3], [1, 2, 3], [0], [1], [0, 1], [0, 1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0035/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 36. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 2, 0, 0], target events left=[2], right=[0, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, 0, 0], [2], [0, 3], [0, 2, 3], [0, 1], [2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0036/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 37. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 2, 0, 0], target events left=[3], right=[0, 1], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 0, 0], [3], [0, 1], [0, 1, 3], [0], [2, 3], [0, 2, 3], [0, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0037/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 38. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 3, 0, 0], target events left=[0], right=[1, 2], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 3, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [], [0, 2, 3], [0, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0038/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 39. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 3, 0, 0], target events left=[1], right=[2, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 3, 0, 0], [1], [2, 3], [1, 2, 3], [0], [1], [0, 1], [0, 1], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0039/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 40. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 3, 0, 0], target events left=[2], right=[0, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3, 0, 0], [2], [0, 3], [0, 2, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0040/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 41. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 3, 0, 0], target events left=[3], right=[0, 1], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 3, 0, 0], [3], [0, 1], [0, 1, 3], [0, 1], [2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0041/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 42. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 0, 1, 0], target events left=[0], right=[1, 2], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 1, 0], [0], [1, 2], [0, 1, 2], [0, 1, 3], [2], [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0042/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 43. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 0, 1, 0], target events left=[1], right=[2, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1, 0], [1], [2, 3], [1, 2, 3], [0, 2], [], [0, 2], [0, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0043/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 44. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 0, 1, 0], target events left=[2], right=[0, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 0, 1, 0], [2], [0, 3], [0, 2, 3], [0], [1, 3], [0, 1, 3], [0, 1, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0044/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 45. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 0, 1, 0], target events left=[3], right=[0, 1], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0, 1, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0045/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 46. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 1, 1, 0], target events left=[0], right=[1, 2], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 1, 0], [0], [1, 2], [0, 1, 2], [0, 3], [1, 2], [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0046/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 47. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 1, 1, 0], target events left=[1], right=[2, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 1, 0], [1], [2, 3], [1, 2, 3], [0, 1, 2], [], [0, 1, 2], [0, 1, 2], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0047/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 48. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[2, 1, 1, 0], target events left=[2], right=[0, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 1, 1, 0], [2], [0, 3], [0, 2, 3], [0], [3], [0, 3], [0, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0048/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 49. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[3, 1, 1, 0], target events left=[3], right=[0, 1], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 1, 1, 0], [3], [0, 1], [0, 1, 3], [0], [1, 2, 3], [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0049/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 50. The Preimage of a Union Equals the Union of Preimages

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[0, 2, 1, 0], target events left=[0], right=[1, 2], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit.

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

Use the fixed total mapping in both constructions and compare sorted source-label sets; no forward-image law is substituted. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2, 1, 0], [0], [1, 2], [0, 1, 2], [0, 3], [1, 2], [0, 1, 2, 3], [0, 1, 2, 3], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0050/)

Measurable function · qf-mod25-measdeep-preimage-union-law · candidate





## 51. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0], [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]], constant target value=0. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 0, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 0, [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-20261025-r05-measurable-function-0051/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 52. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [1], [0, 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]], constant target value=1. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 1, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 1, [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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0052/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 53. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 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]], constant target value=2. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 2, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 2, [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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0053/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 54. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [2], [0, 1, 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]], constant target value=3. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

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

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0054/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 55. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 2], [1, 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]], constant target value=0. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 0, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 0, [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-20261025-r05-measurable-function-0055/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 56. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [1, 2], [0, 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]], constant target value=1. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 1, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 1, [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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0056/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 57. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 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]], constant target value=2. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 2, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 2, [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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0057/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 58. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0], [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]], constant target value=3. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

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

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0058/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 59. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [1], [0, 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]], constant target value=0. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 0, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 0, [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-20261025-r05-measurable-function-0059/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 60. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 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]], constant target value=1. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 1, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 1, [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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0060/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 61. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [2], [0, 1, 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]], constant target value=2. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 2, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 2, [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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0061/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 62. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 2], [1, 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]], constant target value=3. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

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

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0062/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 63. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [1, 2], [0, 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]], constant target value=0. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 0, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 0, [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-20261025-r05-measurable-function-0063/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 64. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 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]], constant target value=1. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 1, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 1, [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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0064/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 65. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0], [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]], constant target value=2. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 2, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 2, [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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0065/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 66. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [1], [0, 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]], constant target value=3. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

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

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0066/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 67. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 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]], constant target value=0. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 0, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 0, [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-20261025-r05-measurable-function-0067/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 68. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [2], [0, 1, 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]], constant target value=1. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 1, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 1, [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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0068/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 69. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 2], [1, 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]], constant target value=2. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 2, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 2, [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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0069/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 70. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [1, 2], [0, 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]], constant target value=3. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

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

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0070/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 71. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 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]], constant target value=0. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 0, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 0, [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-20261025-r05-measurable-function-0071/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 72. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0], [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]], constant target value=1. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 1, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 1, [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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0072/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 73. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [1], [0, 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]], constant target value=2. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 2, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 2, [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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0073/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 74. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [0, 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]], constant target value=3. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

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

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 3, [3, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [], Yes], [[0, 1, 2], [], Yes], [[3], [0, 1, 2, 3], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0074/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 75. All Preimages of a Qualified Constant Finite Map

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, source sigma=[[], [2], [0, 1, 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]], constant target value=0. Give (source size,target size,value,complete constant map,every [target measurable set,preimage,source-measurable?] row,measurable?).

View answer and explanation(4, 4, 0, [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)

Every preimage is either the whole source or the empty set; use the qualified sigma-algebras and retain every target-family row. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, 0, [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-20261025-r05-measurable-function-0075/)

Measurable function · qf-mod25-measdeep-constant-map-audit · candidate





## 76. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[0, 0, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 0, 0], Yes, [[[], [], 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], [[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-20261025-r05-measurable-function-0076/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 77. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[1, 0, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 0, 0], Yes, [[[], [], Yes], [[0], [1, 2, 3], No], [[1], [0], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 2, 3], No], [[1, 2], [0], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], No], [[1, 3], [0], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 2, 3], No], [[1, 2, 3], [0], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], 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-20261025-r05-measurable-function-0077/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 78. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[2, 0, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 0, 0, 0], Yes, [[[], [], 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, [[[], [], Yes], [[0], [1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [0], Yes], [[0, 2], [0, 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], [], Yes], [[0, 1, 3], [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-20261025-r05-measurable-function-0078/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 79. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[3, 0, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0, 0, 0], Yes, [[[], [], Yes], [[0], [1, 2, 3], No], [[1], [], Yes], [[0, 1], [1, 2, 3], No], [[2], [], Yes], [[0, 2], [1, 2, 3], No], [[1, 2], [], Yes], [[0, 1, 2], [1, 2, 3], No], [[3], [0], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[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, [[[], [], 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-20261025-r05-measurable-function-0079/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 80. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[0, 1, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 0, 0], Yes, [[[], [], 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, [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [1], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 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], [1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 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-20261025-r05-measurable-function-0080/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 81. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [1, 2], [0, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[1, 1, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [0, 1], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [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], [0, 1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [2, 3], No], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], 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-20261025-r05-measurable-function-0081/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 82. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], proposed refined 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]], mapping=[2, 1, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 1, 0, 0], Yes, [[[], [], 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, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [1], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [0], Yes], [[0, 2], [0, 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], [1], Yes], [[0, 1, 3], [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-20261025-r05-measurable-function-0082/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 83. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[3, 1, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 1, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [1], No], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], No], [[1, 2], [1], No], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 2, 3], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], 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-20261025-r05-measurable-function-0083/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 84. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[0, 2, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2, 0, 0], Yes, [[[], [], 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, [[[], [], 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-20261025-r05-measurable-function-0084/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 85. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[1, 2, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 2, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0], No], [[0, 1], [0, 2, 3], No], [[2], [1], No], [[0, 2], [1, 2, 3], No], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [0], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1], No], [[0, 2, 3], [1, 2, 3], No], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], 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-20261025-r05-measurable-function-0085/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 86. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[2, 2, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, 0, 0], Yes, [[[], [], 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, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [0, 1], Yes], [[0, 2], [0, 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], [], Yes], [[0, 1, 3], [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-20261025-r05-measurable-function-0086/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 87. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[3, 2, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 2, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [1], No], [[0, 2], [1, 2, 3], No], [[1, 2], [1], No], [[0, 1, 2], [1, 2, 3], No], [[3], [0], No], [[0, 3], [0, 2, 3], No], [[1, 3], [0], No], [[0, 1, 3], [0, 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, [[[], [], 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-20261025-r05-measurable-function-0087/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 88. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [1, 2], [0, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[0, 3, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 3, 0, 0], Yes, [[[], [], 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, [[[], [], 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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261025-r05-measurable-function-0088/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 89. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], proposed refined 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]], mapping=[1, 3, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 3, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [0], No], [[0, 1], [0, 2, 3], No], [[2], [], Yes], [[0, 2], [2, 3], No], [[1, 2], [0], No], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 2, 3], No], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], No], [[0, 2, 3], [1, 2, 3], No], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], 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-20261025-r05-measurable-function-0089/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 90. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[2, 3, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 3, 0, 0], Yes, [[[], [], 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, [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [0], Yes], [[0, 2], [0, 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], [1], Yes], [[0, 1, 3], [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-20261025-r05-measurable-function-0090/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 91. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[3, 3, 0, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 3, 0, 0], Yes, [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [], Yes], [[0, 2], [2, 3], No], [[1, 2], [], Yes], [[0, 1, 2], [2, 3], No], [[3], [0, 1], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1], No], [[0, 1, 3], [0, 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, [[[], [], 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-20261025-r05-measurable-function-0091/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 92. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[0, 0, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 1, 0], Yes, [[[], [], 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, [[[], [], Yes], [[0], [0, 1, 3], Yes], [[1], [2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 3], Yes], [[1, 2], [2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 3], Yes], [[1, 3], [2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [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-20261025-r05-measurable-function-0092/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 93. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[1, 0, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 0, 1, 0], Yes, [[[], [], Yes], [[0], [1, 3], No], [[1], [0, 2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 3], No], [[1, 2], [0, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], No], [[1, 3], [0, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 3], No], [[1, 2, 3], [0, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], 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-20261025-r05-measurable-function-0093/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 94. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[2, 0, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 0, 1, 0], Yes, [[[], [], 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, [[[], [], Yes], [[0], [1, 3], Yes], [[1], [2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [0], Yes], [[0, 2], [0, 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], [2], Yes], [[0, 1, 3], [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-20261025-r05-measurable-function-0094/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 95. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [1, 2], [0, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[3, 0, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 0, 1, 0], Yes, [[[], [], Yes], [[0], [1, 3], No], [[1], [2], No], [[0, 1], [1, 2, 3], No], [[2], [], Yes], [[0, 2], [1, 3], No], [[1, 2], [2], No], [[0, 1, 2], [1, 2, 3], No], [[3], [0], No], [[0, 3], [0, 1, 3], No], [[1, 3], [0, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], 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-20261025-r05-measurable-function-0095/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 96. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], proposed refined 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]], mapping=[0, 1, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 1, 1, 0], Yes, [[[], [], 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, [[[], [], Yes], [[0], [0, 3], Yes], [[1], [1, 2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], Yes], [[1, 2], [1, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], Yes], [[1, 3], [1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 3], Yes], [[1, 2, 3], [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-20261025-r05-measurable-function-0096/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 97. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[1, 1, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 1, 0], Yes, [[[], [], Yes], [[0], [3], No], [[1], [0, 1, 2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [3], No], [[1, 2], [0, 1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], No], [[1, 3], [0, 1, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, [[[], [], 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-20261025-r05-measurable-function-0097/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 98. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[2, 1, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 1, 1, 0], Yes, [[[], [], 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, [[[], [], Yes], [[0], [3], Yes], [[1], [1, 2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [0], Yes], [[0, 2], [0, 3], Yes], [[1, 2], [0, 1, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], Yes], [[1, 3], [1, 2], Yes], [[0, 1, 3], [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-20261025-r05-measurable-function-0098/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 99. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[3, 1, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 1, 1, 0], Yes, [[[], [], Yes], [[0], [3], No], [[1], [1, 2], No], [[0, 1], [1, 2, 3], No], [[2], [], Yes], [[0, 2], [3], No], [[1, 2], [1, 2], No], [[0, 1, 2], [1, 2, 3], No], [[3], [0], No], [[0, 3], [0, 3], No], [[1, 3], [0, 1, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[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, [[[], [], 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-20261025-r05-measurable-function-0099/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate





## 100. Verify Refinement Rather than Assuming the Proposed Family Is Larger

Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, original source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], proposed refined 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]], mapping=[0, 2, 1, 0]. Give (mapping,original family contained in proposed family?,before preimage rows,before measurable?,after rows,after measurable?). Both source families must independently qualify as sigma-algebras.

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

First record whether inclusion really holds, then audit the same mapping in each source structure. A failed inclusion flag cannot be used as a refinement-preservation premise. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 2, 1, 0], Yes, [[[], [], 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, [[[], [], Yes], [[0], [0, 3], Yes], [[1], [2], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [0, 1, 3], Yes], [[1, 2], [1, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], Yes], [[1, 3], [2], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [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-20261025-r05-measurable-function-0100/)

Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261025-r05-measurable-function-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261025-r05-measurable-function-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261025-r05-measurable-function-foundations/validation-report.json)