Finite Preimage Laws, Constant Maps and Source-Sigma Refinement
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Measurable function · qf-mod25-measdeep-source-sigma-refinement-audit · candidate
Canonical question JSONL · English JSONL · Validation report