# Question 11453: Audit Every Target Measurable Set by Preimage [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-0028) 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. R0021-11453 · q-20261013-advance-20261024-r02-measurable-function-0028 ## 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=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?). - 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=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?). ## 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 given map source:carrier Fin(4);complete canonical sigma family=[[], [0, 1], [2, 3], [0, 1, 2, 3]] 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 given map target:carrier Fin(4);complete canonical sigma family=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]] 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, S001 S003 · impliesBefore given map:mapping=[2, 0, 0, 0];source Fin(4);target Fin(4) After complete total map qualified;preimages,not forward images Conditions: none Every source label has one permitted target value.Measurability checks that inverse images of target measurable sets belong to the source sigma-algebra. Step identifiers and dependenciesRule: measurable_total_map_and_direction · Uses: G001, S002 S004 · impliesBefore given map:target event index=0;event=[];mapping=[2, 0, 0, 0] After source membership tests=[No, No, No, No];preimage=[];in source sigma=Yes Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S003 S005 · impliesBefore given map:target event index=1;event=[0];mapping=[2, 0, 0, 0] After source membership tests=[No, Yes, Yes, Yes];preimage=[1, 2, 3];in source sigma=No Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S004 S006 · impliesBefore given map:target event index=2;event=[1];mapping=[2, 0, 0, 0] After source membership tests=[No, No, No, No];preimage=[];in source sigma=Yes Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S005 S007 · impliesBefore given map:target event index=3;event=[0, 1];mapping=[2, 0, 0, 0] After source membership tests=[No, Yes, Yes, Yes];preimage=[1, 2, 3];in source sigma=No Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S006 S008 · impliesBefore given map:target event index=4;event=[2];mapping=[2, 0, 0, 0] After source membership tests=[Yes, No, No, No];preimage=[0];in source sigma=No Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S007 S009 · impliesBefore given map:target event index=5;event=[0, 2];mapping=[2, 0, 0, 0] After source membership tests=[Yes, Yes, Yes, Yes];preimage=[0, 1, 2, 3];in source sigma=Yes Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S008 S010 · impliesBefore given map:target event index=6;event=[1, 2];mapping=[2, 0, 0, 0] After source membership tests=[Yes, No, No, No];preimage=[0];in source sigma=No Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S009 S011 · impliesBefore given map:target event index=7;event=[0, 1, 2];mapping=[2, 0, 0, 0] After source membership tests=[Yes, Yes, Yes, Yes];preimage=[0, 1, 2, 3];in source sigma=Yes Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S010 S012 · impliesBefore given map:target event index=8;event=[3];mapping=[2, 0, 0, 0] After source membership tests=[No, No, No, No];preimage=[];in source sigma=Yes Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S011 S013 · impliesBefore given map:target event index=9;event=[0, 3];mapping=[2, 0, 0, 0] After source membership tests=[No, Yes, Yes, Yes];preimage=[1, 2, 3];in source sigma=No Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S012 S014 · impliesBefore given map:target event index=10;event=[1, 3];mapping=[2, 0, 0, 0] After source membership tests=[No, No, No, No];preimage=[];in source sigma=Yes Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S013 S015 · impliesBefore given map:target event index=11;event=[0, 1, 3];mapping=[2, 0, 0, 0] After source membership tests=[No, Yes, Yes, Yes];preimage=[1, 2, 3];in source sigma=No Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S014 S016 · impliesBefore given map:target event index=12;event=[2, 3];mapping=[2, 0, 0, 0] After source membership tests=[Yes, No, No, No];preimage=[0];in source sigma=No Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S015 S017 · impliesBefore given map:target event index=13;event=[0, 2, 3];mapping=[2, 0, 0, 0] After source membership tests=[Yes, Yes, Yes, Yes];preimage=[0, 1, 2, 3];in source sigma=Yes Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S016 S018 · impliesBefore given map:target event index=14;event=[1, 2, 3];mapping=[2, 0, 0, 0] After source membership tests=[Yes, No, No, No];preimage=[0];in source sigma=No Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S017 S019 · impliesBefore given map:target event index=15;event=[0, 1, 2, 3];mapping=[2, 0, 0, 0] After source membership tests=[Yes, Yes, Yes, Yes];preimage=[0, 1, 2, 3];in source sigma=Yes Conditions: none For each source label test whether f(x) lies in this target event.Collect all passing labels and test the complete source family;do not inspect only singletons or nonempty events. Step identifiers and dependenciesRule: every_target_event_preimage_membership · Uses: G001, S018 S020 · impliesBefore given map:all target preimage rows=[[[], [], 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]] After measurable=No Conditions: none Every target measurable set must pass.One failure is an exact counterexample.This complete finite-space check adds no continuity,integrability or measure-preservation claim. Step identifiers and dependenciesRule: measurability_all_target_events_conjunction · Uses: G001, S019 ## Reverse checks - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 (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) 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: 8e7e297603ba9b52702d7f0dcb6ffb27de78acf12ec0244aca78de921193c250 - candidate record SHA-256: eac2a071cbf4317ee7a600a4aa332185f3c0a1d53a8a8c96a4642ef34e70c3bd - 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-0028.json)