# Question 11416: Composition: Individual Checks, Composite Check and Their Conjunction [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-0091) 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. R0021-11416 · q-20261013-advance-20261024-r02-measurable-function-0091 ## 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, 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. - 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, 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. ## 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 first map source: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 S002 · impliesBefore first 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 first map:mapping=[3, 3, 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 first map:target event index=0;event=[];mapping=[3, 3, 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 first map:target event index=1;event=[0];mapping=[3, 3, 0, 0] After source membership tests=[No, No, Yes, Yes];preimage=[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, S004 S006 · impliesBefore first map:target event index=2;event=[1];mapping=[3, 3, 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 first map:target event index=3;event=[0, 1];mapping=[3, 3, 0, 0] After source membership tests=[No, No, Yes, Yes];preimage=[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, S006 S008 · impliesBefore first map:target event index=4;event=[2];mapping=[3, 3, 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, S007 S009 · impliesBefore first map:target event index=5;event=[0, 2];mapping=[3, 3, 0, 0] After source membership tests=[No, No, Yes, Yes];preimage=[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 first map:target event index=6;event=[1, 2];mapping=[3, 3, 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, S009 S011 · impliesBefore first map:target event index=7;event=[0, 1, 2];mapping=[3, 3, 0, 0] After source membership tests=[No, No, Yes, Yes];preimage=[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 first map:target event index=8;event=[3];mapping=[3, 3, 0, 0] After source membership tests=[Yes, Yes, No, No];preimage=[0, 1];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 first map:target event index=9;event=[0, 3];mapping=[3, 3, 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, S012 S014 · impliesBefore first map:target event index=10;event=[1, 3];mapping=[3, 3, 0, 0] After source membership tests=[Yes, Yes, No, No];preimage=[0, 1];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 first map:target event index=11;event=[0, 1, 3];mapping=[3, 3, 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, S014 S016 · impliesBefore first map:target event index=12;event=[2, 3];mapping=[3, 3, 0, 0] After source membership tests=[Yes, Yes, No, No];preimage=[0, 1];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, S015 S017 · impliesBefore first map:target event index=13;event=[0, 2, 3];mapping=[3, 3, 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 first map:target event index=14;event=[1, 2, 3];mapping=[3, 3, 0, 0] After source membership tests=[Yes, Yes, No, No];preimage=[0, 1];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, S017 S019 · impliesBefore first map:target event index=15;event=[0, 1, 2, 3];mapping=[3, 3, 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 first map:all target preimage rows=[[[], [], 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]] After measurable=Yes 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 S021 · impliesBefore second map source: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, S020 S022 · impliesBefore second 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, S021 S023 · impliesBefore second map:mapping=[0, 1, 3, 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, S022 S024 · impliesBefore second map:target event index=0;event=[];mapping=[0, 1, 3, 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, S023 S025 · impliesBefore second map:target event index=1;event=[0];mapping=[0, 1, 3, 0] After source membership tests=[Yes, No, No, Yes];preimage=[0, 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, S024 S026 · impliesBefore second map:target event index=2;event=[1];mapping=[0, 1, 3, 0] After source membership tests=[No, Yes, No, No];preimage=[1];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, S025 S027 · impliesBefore second map:target event index=3;event=[0, 1];mapping=[0, 1, 3, 0] After source membership tests=[Yes, Yes, No, Yes];preimage=[0, 1, 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, S026 S028 · impliesBefore second map:target event index=4;event=[2];mapping=[0, 1, 3, 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, S027 S029 · impliesBefore second map:target event index=5;event=[0, 2];mapping=[0, 1, 3, 0] After source membership tests=[Yes, No, No, Yes];preimage=[0, 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, S028 S030 · impliesBefore second map:target event index=6;event=[1, 2];mapping=[0, 1, 3, 0] After source membership tests=[No, Yes, No, No];preimage=[1];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, S029 S031 · impliesBefore second map:target event index=7;event=[0, 1, 2];mapping=[0, 1, 3, 0] After source membership tests=[Yes, Yes, No, Yes];preimage=[0, 1, 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, S030 S032 · impliesBefore second map:target event index=8;event=[3];mapping=[0, 1, 3, 0] After source membership tests=[No, No, Yes, No];preimage=[2];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, S031 S033 · impliesBefore second map:target event index=9;event=[0, 3];mapping=[0, 1, 3, 0] After source membership tests=[Yes, No, Yes, Yes];preimage=[0, 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, S032 S034 · impliesBefore second map:target event index=10;event=[1, 3];mapping=[0, 1, 3, 0] After source membership tests=[No, Yes, Yes, No];preimage=[1, 2];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, S033 S035 · impliesBefore second map:target event index=11;event=[0, 1, 3];mapping=[0, 1, 3, 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, S034 S036 · impliesBefore second map:target event index=12;event=[2, 3];mapping=[0, 1, 3, 0] After source membership tests=[No, No, Yes, No];preimage=[2];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, S035 S037 · impliesBefore second map:target event index=13;event=[0, 2, 3];mapping=[0, 1, 3, 0] After source membership tests=[Yes, No, Yes, Yes];preimage=[0, 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, S036 S038 · impliesBefore second map:target event index=14;event=[1, 2, 3];mapping=[0, 1, 3, 0] After source membership tests=[No, Yes, Yes, No];preimage=[1, 2];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, S037 S039 · impliesBefore second map:target event index=15;event=[0, 1, 2, 3];mapping=[0, 1, 3, 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, S038 S040 · impliesBefore second map:all target preimage rows=[[[], [], Yes], [[0], [0, 3], Yes], [[1], [1], Yes], [[0, 1], [0, 1, 3], Yes], [[2], [], Yes], [[0, 2], [0, 3], Yes], [[1, 2], [1], Yes], [[0, 1, 2], [0, 1, 3], Yes], [[3], [2], Yes], [[0, 3], [0, 2, 3], Yes], [[1, 3], [1, 2], Yes], [[0, 1, 3], [0, 1, 2, 3], Yes], [[2, 3], [2], Yes], [[0, 2, 3], [0, 2, 3], Yes], [[1, 2, 3], [1, 2], Yes], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]] After measurable=Yes 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, S039 S041 · impliesBefore source label=0;first image=3;second table=[0, 1, 3, 0] After second-after-first image=0 Conditions: none Apply first to get the middle label,then use it as the index into second.Do not multiply the tables pointwise or reverse composition. Step identifiers and dependenciesRule: composition_declared_order_lookup · Uses: G001, S040 S042 · impliesBefore source label=1;first image=3;second table=[0, 1, 3, 0] After second-after-first image=0 Conditions: none Apply first to get the middle label,then use it as the index into second.Do not multiply the tables pointwise or reverse composition. Step identifiers and dependenciesRule: composition_declared_order_lookup · Uses: G001, S041 S043 · impliesBefore source label=2;first image=0;second table=[0, 1, 3, 0] After second-after-first image=0 Conditions: none Apply first to get the middle label,then use it as the index into second.Do not multiply the tables pointwise or reverse composition. Step identifiers and dependenciesRule: composition_declared_order_lookup · Uses: G001, S042 S044 · impliesBefore source label=3;first image=0;second table=[0, 1, 3, 0] After second-after-first image=0 Conditions: none Apply first to get the middle label,then use it as the index into second.Do not multiply the tables pointwise or reverse composition. Step identifiers and dependenciesRule: composition_declared_order_lookup · Uses: G001, S043 S045 · impliesBefore composite map source: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, S044 S046 · impliesBefore composite 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, S045 S047 · impliesBefore composite map:mapping=[0, 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, S046 S048 · impliesBefore composite map:target event index=0;event=[];mapping=[0, 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, S047 S049 · impliesBefore composite map:target event index=1;event=[0];mapping=[0, 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, S048 S050 · impliesBefore composite map:target event index=2;event=[1];mapping=[0, 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, S049 S051 · impliesBefore composite map:target event index=3;event=[0, 1];mapping=[0, 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, S050 S052 · impliesBefore composite map:target event index=4;event=[2];mapping=[0, 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, S051 S053 · impliesBefore composite map:target event index=5;event=[0, 2];mapping=[0, 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, S052 S054 · impliesBefore composite map:target event index=6;event=[1, 2];mapping=[0, 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, S053 S055 · impliesBefore composite map:target event index=7;event=[0, 1, 2];mapping=[0, 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, S054 S056 · impliesBefore composite map:target event index=8;event=[3];mapping=[0, 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, S055 S057 · impliesBefore composite map:target event index=9;event=[0, 3];mapping=[0, 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, S056 S058 · impliesBefore composite map:target event index=10;event=[1, 3];mapping=[0, 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, S057 S059 · impliesBefore composite map:target event index=11;event=[0, 1, 3];mapping=[0, 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, S058 S060 · impliesBefore composite map:target event index=12;event=[2, 3];mapping=[0, 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, S059 S061 · impliesBefore composite map:target event index=13;event=[0, 2, 3];mapping=[0, 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, S060 S062 · impliesBefore composite map:target event index=14;event=[1, 2, 3];mapping=[0, 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, S061 S063 · impliesBefore composite map:target event index=15;event=[0, 1, 2, 3];mapping=[0, 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, S062 S064 · impliesBefore composite map:all target preimage rows=[[[], [], 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]] After measurable=Yes 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, S063 S065 · impliesBefore first measurable=Yes;second measurable=Yes;composite measurable=Yes After final source flag is conjunction=Yes,not implication Conditions: none The identity(g∘f)^−1(E)=f^−1(g^−1(E)) proves the theorem when both factors are measurable.The requested last field is all three checks together,not the truth value of that implication.A measurable composite need not have two measurable factors. Step identifiers and dependenciesRule: composition_conjunction_not_theorem_implication · Uses: G001, S064 ## Reverse checks - S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S052 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 (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) 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: 66bb0fdeab577aefe98741cf532f5b38164f00b296e33ab630def61d5953ccf9 - candidate record SHA-256: e6536325d0933dc1eb3059f15ba88240084c66805f41b10cc4429b04bd0feaf4 - 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-0091.json)