# Question 11429: An Indicator’s Measurability Depends on Its Event [Return to catalogue page 58](https://ocme.evemisslab.com/en/questions/page/58/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261024-r02-measurable-function-foundations/#q-20261013-advance-20261024-r02-measurable-function-0054) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0021-11429 · q-20261013-advance-20261024-r02-measurable-function-0054 ## Original question and givens 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?). - G001 — 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?). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-measurable-function](https://ocme.evemisslab.com/en/mko/mko-measurable-function/) ## Step-by-step derivation S001 · impliesBefore indicator source:carrier Fin(6);complete canonical sigma family=[[], [2], [0, 1, 3, 4, 5], [0, 1, 2, 3, 4, 5]] After empty member=Yes;whole member=Yes;missing complements=[];missing pair unions=[] Conditions: none Qualify the complete bit-mask-ordered family:empty set,universe,complements and all pair unions.In a finite family a countable union has only finitely many distinct members,so finite-union closure suffices. Step identifiers and dependenciesRule: qualify_complete_finite_sigma_algebra · Uses: G001 S002 · impliesBefore source label=0;normalized event=[0] After indicator value=1 Conditions: none Event members map to1;other source points map to0.The indicator's target is{0,1},not the source carrier. Step identifiers and dependenciesRule: indicator_one_on_event_zero_elsewhere · Uses: G001, S001 S003 · impliesBefore source label=1;normalized event=[0] After indicator value=0 Conditions: none Event members map to1;other source points map to0.The indicator's target is{0,1},not the source carrier. Step identifiers and dependenciesRule: indicator_one_on_event_zero_elsewhere · Uses: G001, S002 S004 · impliesBefore source label=2;normalized event=[0] After indicator value=0 Conditions: none Event members map to1;other source points map to0.The indicator's target is{0,1},not the source carrier. Step identifiers and dependenciesRule: indicator_one_on_event_zero_elsewhere · Uses: G001, S003 S005 · impliesBefore source label=3;normalized event=[0] After indicator value=0 Conditions: none Event members map to1;other source points map to0.The indicator's target is{0,1},not the source carrier. Step identifiers and dependenciesRule: indicator_one_on_event_zero_elsewhere · Uses: G001, S004 S006 · impliesBefore source label=4;normalized event=[0] After indicator value=0 Conditions: none Event members map to1;other source points map to0.The indicator's target is{0,1},not the source carrier. Step identifiers and dependenciesRule: indicator_one_on_event_zero_elsewhere · Uses: G001, S005 S007 · impliesBefore source label=5;normalized event=[0] After indicator value=0 Conditions: none Event members map to1;other source points map to0.The indicator's target is{0,1},not the source carrier. Step identifiers and dependenciesRule: indicator_one_on_event_zero_elsewhere · Uses: G001, S006 S008 · impliesBefore event=[0];indicator mapping=[1, 0, 0, 0, 0, 0];target full sigma=[[],[0],[1],[0,1]] After preimages=[[],[1, 2, 3, 4, 5],[0],[0, 1, 2, 3, 4, 5]];event in source sigma=No Conditions: none The four target preimages are empty,event complement,event,and universe.The qualified source sigma contains empty/universe and is complement-closed,so indicator measurability is equivalent to event measurability. Step identifiers and dependenciesRule: indicator_all_four_preimages_equivalence · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (6, [0], [1, 0, 0, 0, 0, 0], No) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 06d46d5dc918d2b0ff46258c452914594b160ceea1db8509da110b00ea2616b5 - candidate record SHA-256: 6e4285d63e97b9322743d96216a2cf4a073f37a7a6a5c6f729677878af8cae1d - semantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258 - question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261024-r02-measurable-function-0054.json)