Complete Finite Measurability through Target Preimages
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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-20261024-r02-measurable-function-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[0, 0, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [0, 0, 0, 0], [0, 1], [0, 1, 2, 3])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 0, 0, 0], [0, 1], [0, 1, 2, 3]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
2. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[1, 0, 0, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [1, 0, 0, 0], [1, 2], [0])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 0, 0, 0], [1, 2], [0]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
3. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[2, 0, 0, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [2, 0, 0, 0], [2, 3], [0])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 0, 0, 0], [2, 3], [0]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
4. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[3, 0, 0, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [3, 0, 0, 0], [0, 3], [0, 1, 2, 3])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 0, 0, 0], [0, 3], [0, 1, 2, 3]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
5. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[0, 1, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [0, 1, 0, 0], [0, 1], [0, 1, 2, 3])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 1, 0, 0], [0, 1], [0, 1, 2, 3]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
6. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[1, 1, 0, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [1, 1, 0, 0], [1, 2], [0, 1])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 1, 0, 0], [1, 2], [0, 1]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
7. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[2, 1, 0, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [2, 1, 0, 0], [2, 3], [0])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 1, 0, 0], [2, 3], [0]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
8. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[3, 1, 0, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [3, 1, 0, 0], [0, 3], [0, 2, 3])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 1, 0, 0], [0, 3], [0, 2, 3]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
9. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[0, 2, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [0, 2, 0, 0], [0, 1], [0, 2, 3])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 2, 0, 0], [0, 1], [0, 2, 3]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
10. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[1, 2, 0, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [1, 2, 0, 0], [1, 2], [0, 1])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 2, 0, 0], [1, 2], [0, 1]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
11. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[2, 2, 0, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [2, 2, 0, 0], [2, 3], [0, 1])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 2, 0, 0], [2, 3], [0, 1]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
12. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[3, 2, 0, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [3, 2, 0, 0], [0, 3], [0, 2, 3])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 2, 0, 0], [0, 3], [0, 2, 3]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
13. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[0, 3, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [0, 3, 0, 0], [0, 1], [0, 2, 3])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 3, 0, 0], [0, 1], [0, 2, 3]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
14. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[1, 3, 0, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [1, 3, 0, 0], [1, 2], [0])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 3, 0, 0], [1, 2], [0]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
15. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[2, 3, 0, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [2, 3, 0, 0], [2, 3], [0, 1])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 3, 0, 0], [2, 3], [0, 1]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
16. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[3, 3, 0, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [3, 3, 0, 0], [0, 3], [0, 1, 2, 3])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 3, 0, 0], [0, 3], [0, 1, 2, 3]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
17. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[0, 0, 1, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [0, 0, 1, 0], [0, 1], [0, 1, 2, 3])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 0, 1, 0], [0, 1], [0, 1, 2, 3]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
18. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[1, 0, 1, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [1, 0, 1, 0], [1, 2], [0, 2])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 0, 1, 0], [1, 2], [0, 2]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
19. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[2, 0, 1, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [2, 0, 1, 0], [2, 3], [0])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 0, 1, 0], [2, 3], [0]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
20. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[3, 0, 1, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [3, 0, 1, 0], [0, 3], [0, 1, 3])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 0, 1, 0], [0, 3], [0, 1, 3]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
21. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[0, 1, 1, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [0, 1, 1, 0], [0, 1], [0, 1, 2, 3])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 1, 1, 0], [0, 1], [0, 1, 2, 3]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
22. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[1, 1, 1, 0] indexed by source labels, and target event=[1, 2] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [1, 1, 1, 0], [1, 2], [0, 1, 2])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 1, 1, 0], [1, 2], [0, 1, 2]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
23. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[2, 1, 1, 0] indexed by source labels, and target event=[2, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [2, 1, 1, 0], [2, 3], [0])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 1, 1, 0], [2, 3], [0]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
24. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[3, 1, 1, 0] indexed by source labels, and target event=[0, 3] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [3, 1, 1, 0], [0, 3], [0, 3])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 1, 1, 0], [0, 3], [0, 3]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
25. The Inverse Image under a Complete Finite Total Map
Use carriers Fin(4) and Fin(4), complete total mapping=[0, 2, 1, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.
View answer and explanation
(4, 4, [0, 2, 1, 0], [0, 1], [0, 2, 3])
Filter the complete source carrier using target-event membership, preserving source-label order and distinguishing inverse images from forward images. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 2, 1, 0], [0, 1], [0, 2, 3]).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-preimage · candidate
26. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 0, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
27. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 0, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [1, 0, 0, 0], [[[], [], Yes], [[0], [1, 2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 0, 0, 0], [[[], [], Yes], [[0], [1, 2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
28. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 0, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [2, 0, 0, 0], [[[], [], Yes], [[0], [1, 2, 3], No], [[1], [], Yes], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], No], [[1, 3], [], Yes], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 0, 0, 0], [[[], [], Yes], [[0], [1, 2, 3], No], [[1], [], Yes], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 2, 3], No], [[1, 3], [], Yes], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
29. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 0, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [3, 0, 0, 0], [[[], [], Yes], [[0], [1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 0, 0, 0], [[[], [], Yes], [[0], [1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
30. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 1, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [0, 1, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], No], [[1], [1], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], No], [[1, 3], [1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 1, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], No], [[1], [1], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], No], [[1, 3], [1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
31. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 1, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [1, 1, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0, 1], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 1, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0, 1], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
32. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 1, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [2, 1, 0, 0], [[[], [], Yes], [[0], [2, 3], No], [[1], [1], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 2, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], No], [[1, 3], [1], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 1, 0, 0], [[[], [], Yes], [[0], [2, 3], No], [[1], [1], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 2, 3], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], No], [[1, 3], [1], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 2, 3], No], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
33. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 1, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [3, 1, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [1], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 1, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [1], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
34. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 2, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [0, 2, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 2, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
35. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 2, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [1, 2, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 2, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0, 1], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
36. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 2, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [2, 2, 0, 0], [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [0, 1], No], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], No], [[1, 3], [], Yes], [[0, 1, 3], [2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 2, 0, 0], [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [0, 1], No], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [2, 3], No], [[1, 3], [], Yes], [[0, 1, 3], [2, 3], No], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
37. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 2, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [3, 2, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [1], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 2, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [1], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
38. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [1, 2], [0, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 3, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [0, 3, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], No], [[1], [], Yes], [[0, 1], [0, 2, 3], No], [[2], [], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 3, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], No], [[1], [], Yes], [[0, 1], [0, 2, 3], No], [[2], [], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
39. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 3, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [1, 3, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 3, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [0], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [0], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [1, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
40. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 3, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [2, 3, 0, 0], [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [0], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [0], Yes], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 2, 3], Yes], [[1, 3], [1], No], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 3, 0, 0], [[[], [], Yes], [[0], [2, 3], No], [[1], [], Yes], [[0, 1], [2, 3], No], [[2], [0], Yes], [[0, 2], [0, 2, 3], No], [[1, 2], [0], Yes], [[0, 1, 2], [0, 2, 3], No], [[3], [1], No], [[0, 3], [1, 2, 3], Yes], [[1, 3], [1], No], [[0, 1, 3], [1, 2, 3], Yes], [[2, 3], [0, 1], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
41. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 3, 0, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [3, 3, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [2, 3], Yes], [[3], [0, 1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 3, 0, 0], [[[], [], Yes], [[0], [2, 3], Yes], [[1], [], Yes], [[0, 1], [2, 3], Yes], [[2], [], Yes], [[0, 2], [2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [2, 3], Yes], [[3], [0, 1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
42. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 0, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [0, 0, 1, 0], [[[], [], Yes], [[0], [0, 1, 3], No], [[1], [2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 3], No], [[1, 2], [2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 3], No], [[1, 3], [2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 0, 1, 0], [[[], [], Yes], [[0], [0, 1, 3], No], [[1], [2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 3], No], [[1, 2], [2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 3], No], [[1, 3], [2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
43. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 0, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [1, 0, 1, 0], [[[], [], Yes], [[0], [1, 3], Yes], [[1], [0, 2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 3], Yes], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 0, 1, 0], [[[], [], Yes], [[0], [1, 3], Yes], [[1], [0, 2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1, 3], Yes], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
44. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 0, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [2, 0, 1, 0], [[[], [], Yes], [[0], [1, 3], Yes], [[1], [2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 1, 3], No], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], Yes], [[1, 3], [2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 0, 1, 0], [[[], [], Yes], [[0], [1, 3], Yes], [[1], [2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 1, 3], No], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1, 3], Yes], [[1, 3], [2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
45. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 0, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [3, 0, 1, 0], [[[], [], Yes], [[0], [1, 3], Yes], [[1], [2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [2], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 1, 3], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 0, 1, 0], [[[], [], Yes], [[0], [1, 3], Yes], [[1], [2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [2], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 1, 3], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [0, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
46. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 1, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [0, 1, 1, 0], [[[], [], Yes], [[0], [0, 3], No], [[1], [1, 2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 1, 1, 0], [[[], [], Yes], [[0], [0, 3], No], [[1], [1, 2], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], No], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
47. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[1, 1, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [1, 1, 1, 0], [[[], [], Yes], [[0], [3], Yes], [[1], [0, 1, 2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [3], Yes], [[1, 2], [0, 1, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], Yes], [[1, 3], [0, 1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [3], Yes], [[1, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [1, 1, 1, 0], [[[], [], Yes], [[0], [3], Yes], [[1], [0, 1, 2], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [3], Yes], [[1, 2], [0, 1, 2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], Yes], [[1, 3], [0, 1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [3], Yes], [[1, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
48. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 1, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [2, 1, 1, 0], [[[], [], Yes], [[0], [3], No], [[1], [1, 2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [2, 1, 1, 0], [[[], [], Yes], [[0], [3], No], [[1], [1, 2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
49. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[3, 1, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [3, 1, 1, 0], [[[], [], Yes], [[0], [3], Yes], [[1], [1, 2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [3], Yes], [[1, 2], [1, 2], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 3], Yes], [[1, 3], [0, 1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 3], Yes], [[1, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [3, 1, 1, 0], [[[], [], Yes], [[0], [3], Yes], [[1], [1, 2], Yes], [[0, 1], [1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [3], Yes], [[1, 2], [1, 2], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0, 3], Yes], [[1, 3], [0, 1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0], Yes], [[0, 2, 3], [0, 3], Yes], [[1, 2, 3], [0, 1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
50. Audit Every Target Measurable Set by Preimage
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[0, 2, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).
View answer and explanation
(4, 4, [0, 2, 1, 0], [[[], [], Yes], [[0], [0, 3], No], [[1], [2], Yes], [[0, 1], [0, 2, 3], No], [[2], [1], No], [[0, 2], [0, 1, 3], Yes], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], No], [[1, 3], [2], Yes], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1], No], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)
Enumerate the complete target family in its canonical order. Every preimage must belong to the fixed source family. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (4, 4, [0, 2, 1, 0], [[[], [], Yes], [[0], [0, 3], No], [[1], [2], Yes], [[0, 1], [0, 2, 3], No], [[2], [1], No], [[0, 2], [0, 1, 3], Yes], [[1, 2], [1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 3], No], [[1, 3], [2], Yes], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1], No], [[0, 2, 3], [0, 1, 3], Yes], [[1, 2, 3], [1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-map-audit · candidate
51. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0], [1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0], [1, 0, 0, 0, 0, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
52. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1], [0, 2, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[1] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [1], [0, 1, 0, 0, 0, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [1], [0, 1, 0, 0, 0, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
53. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1], [2, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0, 1] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0, 1], [1, 1, 0, 0, 0, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 1], [1, 1, 0, 0, 0, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
54. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [2], [0, 1, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0], [1, 0, 0, 0, 0, 0], No)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
55. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 2], [1, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0, 2] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0, 2], [1, 0, 1, 0, 0, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 2], [1, 0, 1, 0, 0, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
56. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1, 2], [0, 3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[1, 2] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [1, 2], [0, 1, 1, 0, 0, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [1, 2], [0, 1, 1, 0, 0, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
57. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 2], [3, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0], [1, 0, 0, 0, 0, 0], No)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
58. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [3], [0, 1, 2, 4, 5], [0, 1, 2, 3, 4, 5]], event=[3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [3], [0, 0, 0, 1, 0, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [3], [0, 0, 0, 1, 0, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
59. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 3], [1, 2, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0, 3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0, 3], [1, 0, 0, 1, 0, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 3], [1, 0, 0, 1, 0, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
60. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1, 3], [0, 2, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0], [1, 0, 0, 0, 0, 0], No)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
61. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 3], [2, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0, 1, 3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0, 1, 3], [1, 1, 0, 1, 0, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 1, 3], [1, 1, 0, 1, 0, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
62. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [2, 3], [0, 1, 4, 5], [0, 1, 2, 3, 4, 5]], event=[2, 3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [2, 3], [0, 0, 1, 1, 0, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [2, 3], [0, 0, 1, 1, 0, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
63. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 2, 3], [1, 4, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0], [1, 0, 0, 0, 0, 0], No)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
64. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1, 2, 3], [0, 4, 5], [0, 1, 2, 3, 4, 5]], event=[1, 2, 3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [1, 2, 3], [0, 1, 1, 1, 0, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [1, 2, 3], [0, 1, 1, 1, 0, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
65. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 2, 3], [4, 5], [0, 1, 2, 3, 4, 5]], event=[0, 1, 2, 3] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0, 1, 2, 3], [1, 1, 1, 1, 0, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 1, 2, 3], [1, 1, 1, 1, 0, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
66. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [4], [0, 1, 2, 3, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0], [1, 0, 0, 0, 0, 0], No)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
67. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 4], [1, 2, 3, 5], [0, 1, 2, 3, 4, 5]], event=[0, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0, 4], [1, 0, 0, 0, 1, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 4], [1, 0, 0, 0, 1, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
68. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1, 4], [0, 2, 3, 5], [0, 1, 2, 3, 4, 5]], event=[1, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [1, 4], [0, 1, 0, 0, 1, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [1, 4], [0, 1, 0, 0, 1, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
69. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 4], [2, 3, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0], [1, 0, 0, 0, 0, 0], No)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
70. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [2, 4], [0, 1, 3, 5], [0, 1, 2, 3, 4, 5]], event=[2, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [2, 4], [0, 0, 1, 0, 1, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [2, 4], [0, 0, 1, 0, 1, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
71. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 2, 4], [1, 3, 5], [0, 1, 2, 3, 4, 5]], event=[0, 2, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0, 2, 4], [1, 0, 1, 0, 1, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 2, 4], [1, 0, 1, 0, 1, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
72. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [1, 2, 4], [0, 3, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0], [1, 0, 0, 0, 0, 0], No)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
73. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 1, 2, 4], [3, 5], [0, 1, 2, 3, 4, 5]], event=[0, 1, 2, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0, 1, 2, 4], [1, 1, 1, 0, 1, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0, 1, 2, 4], [1, 1, 1, 0, 1, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
74. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [3, 4], [0, 1, 2, 5], [0, 1, 2, 3, 4, 5]], event=[3, 4] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [3, 4], [0, 0, 0, 1, 1, 0], Yes)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [3, 4], [0, 0, 0, 1, 1, 0], Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
75. An Indicator’s Measurability Depends on Its Event
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=6, source sigma=[[], [0, 3, 4], [1, 2, 5], [0, 1, 2, 3, 4, 5]], event=[0] normalized as a set. The indicator maps members to 1 and others to 0, with the full measurable structure on {0,1}. Give (source size,event,complete indicator mapping,event is source-measurable?).
View answer and explanation
(6, [0], [1, 0, 0, 0, 0, 0], No)
The inverse images of the two singleton target values are the event and its complement. The qualified source sigma-algebra handles both together. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (6, [0], [1, 0, 0, 0, 0, 0], No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-indicator-audit · candidate
76. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[0, 0, 0, 0], second=[1, 1, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(Yes, Yes, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
77. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[1, 0, 0, 0], second=[2, 1, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(Yes, Yes, [1, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [0], Yes], [[0, 1], [0], Yes], [[2], [1, 2, 3], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0], Yes], [[2, 3], [1, 2, 3], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [1, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [0], Yes], [[0, 1], [0], Yes], [[2], [1, 2, 3], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0], Yes], [[2, 3], [1, 2, 3], Yes], [[0, 2, 3], [1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
78. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[2, 0, 0, 0], second=[3, 1, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(No, Yes, [2, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0], No], [[0, 2], [0], No], [[1, 2], [0], No], [[0, 1, 2], [0], No], [[3], [1, 2, 3], No], [[0, 3], [1, 2, 3], No], [[1, 3], [1, 2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [2, 3, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0], No], [[0, 2], [0], No], [[1, 2], [0], No], [[0, 1, 2], [0], No], [[3], [1, 2, 3], No], [[0, 3], [1, 2, 3], No], [[1, 3], [1, 2, 3], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
79. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[3, 0, 0, 0], second=[0, 2, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(Yes, Yes, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
80. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[0, 1, 0, 0], second=[1, 2, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(No, Yes, [1, 2, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2, 3], No], [[0, 1], [0, 2, 3], No], [[2], [1], No], [[0, 2], [1], No], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1], No], [[0, 2, 3], [1], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [1, 2, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2, 3], No], [[0, 1], [0, 2, 3], No], [[2], [1], No], [[0, 2], [1], No], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 2, 3], No], [[2, 3], [1], No], [[0, 2, 3], [1], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
81. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[1, 1, 0, 0], second=[2, 2, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(Yes, Yes, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
82. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[2, 1, 0, 0], second=[3, 2, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(No, Yes, [2, 2, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1], No], [[0, 2], [0, 1], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1], No], [[3], [2, 3], No], [[0, 3], [2, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [2, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [2, 2, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1], No], [[0, 2], [0, 1], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1], No], [[3], [2, 3], No], [[0, 3], [2, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [2, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
83. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[3, 1, 0, 0], second=[0, 3, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(Yes, Yes, [0, 3, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 3, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
84. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[0, 2, 0, 0], second=[1, 3, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(Yes, Yes, [1, 2, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2, 3], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [1], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2, 3], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [1, 2, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2, 3], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [1], Yes], [[0, 2], [1], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2, 3], Yes], [[0, 1, 3], [0, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [1], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
85. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[1, 2, 0, 0], second=[2, 3, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(Yes, Yes, [3, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [1, 2, 3], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [1, 2, 3], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [3, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [1, 2, 3], Yes], [[0, 2], [1, 2, 3], Yes], [[1, 2], [1, 2, 3], Yes], [[0, 1, 2], [1, 2, 3], Yes], [[3], [0], Yes], [[0, 3], [0], Yes], [[1, 3], [0], Yes], [[0, 1, 3], [0], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
86. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[2, 2, 0, 0], second=[3, 3, 2, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(No, Yes, [2, 2, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1], No], [[0, 2], [0, 1], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1], No], [[3], [2, 3], No], [[0, 3], [2, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [2, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [2, 2, 3, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1], No], [[0, 2], [0, 1], No], [[1, 2], [0, 1], No], [[0, 1, 2], [0, 1], No], [[3], [2, 3], No], [[0, 3], [2, 3], No], [[1, 3], [2, 3], No], [[0, 1, 3], [2, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
87. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[3, 2, 0, 0], second=[0, 0, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(Yes, Yes, [0, 3, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 3, 0, 0], [[[], [], Yes], [[0], [0, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
88. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [1, 2], [0, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[0, 3, 0, 0], second=[1, 0, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(No, Yes, [1, 0, 1, 1], [[[], [], Yes], [[0], [1], No], [[1], [0, 2, 3], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1], No], [[1, 2], [0, 2, 3], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1], No], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [1, 0, 1, 1], [[[], [], Yes], [[0], [1], No], [[1], [0, 2, 3], No], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [1], No], [[1, 2], [0, 2, 3], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [1], No], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [1], No], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
89. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[1, 3, 0, 0], second=[2, 0, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(Yes, Yes, [0, 0, 2, 2], [[[], [], Yes], [[0], [0, 1], Yes], [[1], [], Yes], [[0, 1], [0, 1], Yes], [[2], [2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1], Yes], [[2, 3], [2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 0, 2, 2], [[[], [], Yes], [[0], [0, 1], Yes], [[1], [], Yes], [[0, 1], [0, 1], Yes], [[2], [2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1], Yes], [[2, 3], [2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
90. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[2, 3, 0, 0], second=[3, 0, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(No, Yes, [3, 0, 3, 3], [[[], [], Yes], [[0], [1], No], [[1], [], Yes], [[0, 1], [1], No], [[2], [], Yes], [[0, 2], [1], No], [[1, 2], [], Yes], [[0, 1, 2], [1], No], [[3], [0, 2, 3], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 2, 3], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [3, 0, 3, 3], [[[], [], Yes], [[0], [1], No], [[1], [], Yes], [[0, 1], [1], No], [[2], [], Yes], [[0, 2], [1], No], [[1, 2], [], Yes], [[0, 1, 2], [1], No], [[3], [0, 2, 3], No], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [0, 2, 3], No], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 2, 3], No], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 2, 3], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
91. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[3, 3, 0, 0], second=[0, 1, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(Yes, Yes, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 0, 0, 0], [[[], [], Yes], [[0], [0, 1, 2, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
92. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 1], [2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[0, 0, 1, 0], second=[1, 1, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(No, Yes, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, No)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [1, 1, 1, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 1, 2, 3], Yes], [[0, 1], [0, 1, 2, 3], Yes], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [], Yes], [[0, 2, 3], [], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
93. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[1, 0, 1, 0], second=[2, 1, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(Yes, Yes, [1, 2, 1, 2], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2], Yes], [[0, 1], [0, 2], Yes], [[2], [1, 3], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 2], Yes], [[2, 3], [1, 3], Yes], [[0, 2, 3], [1, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [1, 2, 1, 2], [[[], [], Yes], [[0], [], Yes], [[1], [0, 2], Yes], [[0, 1], [0, 2], Yes], [[2], [1, 3], Yes], [[0, 2], [1, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 2], Yes], [[0, 1, 3], [0, 2], Yes], [[2, 3], [1, 3], Yes], [[0, 2, 3], [1, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
94. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 2], [1, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[2, 0, 1, 0], second=[3, 1, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(No, Yes, [3, 3, 1, 3], [[[], [], Yes], [[0], [], Yes], [[1], [2], No], [[0, 1], [2], No], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [2], No], [[0, 1, 2], [2], No], [[3], [0, 1, 3], No], [[0, 3], [0, 1, 3], No], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [3, 3, 1, 3], [[[], [], Yes], [[0], [], Yes], [[1], [2], No], [[0, 1], [2], No], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [2], No], [[0, 1, 2], [2], No], [[3], [0, 1, 3], No], [[0, 3], [0, 1, 3], No], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [0, 1, 3], No], [[0, 2, 3], [0, 1, 3], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
95. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[3, 0, 1, 0], second=[0, 2, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(Yes, Yes, [0, 0, 2, 0], [[[], [], Yes], [[0], [0, 1, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 3], Yes], [[2], [2], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 3], Yes], [[2, 3], [2], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 0, 2, 0], [[[], [], Yes], [[0], [0, 1, 3], Yes], [[1], [], Yes], [[0, 1], [0, 1, 3], Yes], [[2], [2], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [2], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [0, 1, 3], Yes], [[1, 3], [], Yes], [[0, 1, 3], [0, 1, 3], Yes], [[2, 3], [2], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
96. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0, 1, 2], [3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[0, 1, 1, 0], second=[1, 2, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(No, Yes, [1, 2, 2, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 3], No], [[0, 1], [0, 3], No], [[2], [1, 2], No], [[0, 2], [1, 2], No], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 3], No], [[2, 3], [1, 2], No], [[0, 2, 3], [1, 2], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [1, 2, 2, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 3], No], [[0, 1], [0, 3], No], [[2], [1, 2], No], [[0, 2], [1, 2], No], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 3], No], [[2, 3], [1, 2], No], [[0, 2, 3], [1, 2], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
97. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[1, 1, 1, 0], second=[2, 2, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(Yes, Yes, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [2, 2, 2, 2], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [0, 1, 2, 3], Yes], [[0, 2], [0, 1, 2, 3], Yes], [[1, 2], [0, 1, 2, 3], Yes], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [], Yes], [[1, 3], [], Yes], [[0, 1, 3], [], Yes], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
98. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [1], [0, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[2, 1, 1, 0], second=[3, 2, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(No, Yes, [3, 2, 2, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [1, 2], No], [[0, 2], [1, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [1, 2], No], [[3], [0, 3], No], [[0, 3], [0, 3], No], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [3, 2, 2, 3], [[[], [], Yes], [[0], [], Yes], [[1], [], Yes], [[0, 1], [], Yes], [[2], [1, 2], No], [[0, 2], [1, 2], No], [[1, 2], [1, 2], No], [[0, 1, 2], [1, 2], No], [[3], [0, 3], No], [[0, 3], [0, 3], No], [[1, 3], [0, 3], No], [[0, 1, 3], [0, 3], No], [[2, 3], [0, 1, 2, 3], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
99. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[3, 1, 1, 0], second=[0, 3, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(Yes, Yes, [0, 3, 3, 0], [[[], [], Yes], [[0], [0, 3], Yes], [[1], [], Yes], [[0, 1], [0, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 3], Yes], [[3], [1, 2], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1, 2], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (Yes, Yes, [0, 3, 3, 0], [[[], [], Yes], [[0], [0, 3], Yes], [[1], [], Yes], [[0, 1], [0, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], Yes], [[1, 2], [], Yes], [[0, 1, 2], [0, 3], Yes], [[3], [1, 2], Yes], [[0, 3], [0, 1, 2, 3], Yes], [[1, 3], [1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1, 2], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], Yes, Yes).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
100. Composition: Individual Checks, Composite Check and Their Conjunction
Use complete qualified finite source and target sigma-algebras. Each subset has sorted distinct carrier labels; the duplicate-free family is already ordered by increasing bit mask. Check preimages of target measurable sets, not images of source sets. Measurable does not mean continuous, integrable or measure-preserving. source size=4, middle size=4, target size=4; source sigma=[[], [2], [0, 1, 3], [0, 1, 2, 3]], middle sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]]. First map=[0, 2, 1, 0], second=[1, 3, 3, 0]. Give (first measurable?,second measurable?,second-after-first mapping,every composite preimage row,composite measurable?,all three checks pass?). The final source flag is a conjunction, not the implication that two measurable maps compose to a measurable map.
View answer and explanation
(No, Yes, [1, 3, 3, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 3], No], [[0, 1], [0, 3], No], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 3], No], [[3], [1, 2], No], [[0, 3], [1, 2], No], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1, 2], No], [[0, 2, 3], [1, 2], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No)
Compose in the declared order and audit every target preimage. A composite can be measurable even when one factor is not; the final conjunction then remains No without contradicting the composition theorem. This exact finite slice is not automatically a general infinite conclusion. The exact result, retaining the requested order, is (No, Yes, [1, 3, 3, 1], [[[], [], Yes], [[0], [], Yes], [[1], [0, 3], No], [[0, 1], [0, 3], No], [[2], [], Yes], [[0, 2], [], Yes], [[1, 2], [0, 3], No], [[0, 1, 2], [0, 3], No], [[3], [1, 2], No], [[0, 3], [1, 2], No], [[1, 3], [0, 1, 2, 3], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [1, 2], No], [[0, 2, 3], [1, 2], No], [[1, 2, 3], [0, 1, 2, 3], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No, No).
Reviewed step-by-step derivation
Measurable function · qf-mod24-measurable-composition-audit · candidate
Canonical question JSONL · English JSONL · Validation report