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Question 11410: Composition: Individual Checks, Composite Check and Their Conjunction

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0021-11410 · q-20261013-advance-20261024-r02-measurable-function-0085

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=[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.

  • 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=[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.

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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: qualify_complete_finite_sigma_algebra · Uses: G001

S002 · implies

Before

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 dependencies

Rule: qualify_complete_finite_sigma_algebra · Uses: G001, S001

S003 · implies

Before

first map:mapping=[1, 2, 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 dependencies

Rule: measurable_total_map_and_direction · Uses: G001, S002

S004 · implies

Before

first map:target event index=0;event=[];mapping=[1, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S003

S005 · implies

Before

first map:target event index=1;event=[0];mapping=[1, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S004

S006 · implies

Before

first map:target event index=2;event=[1];mapping=[1, 2, 0, 0]

After

source membership tests=[Yes, No, No, No];preimage=[0];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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S005

S007 · implies

Before

first map:target event index=3;event=[0, 1];mapping=[1, 2, 0, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S006

S008 · implies

Before

first map:target event index=4;event=[2];mapping=[1, 2, 0, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S007

S009 · implies

Before

first map:target event index=5;event=[0, 2];mapping=[1, 2, 0, 0]

After

source membership tests=[No, Yes, Yes, Yes];preimage=[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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S008

S010 · implies

Before

first map:target event index=6;event=[1, 2];mapping=[1, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S009

S011 · implies

Before

first map:target event index=7;event=[0, 1, 2];mapping=[1, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S010

S012 · implies

Before

first map:target event index=8;event=[3];mapping=[1, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S011

S013 · implies

Before

first map:target event index=9;event=[0, 3];mapping=[1, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S012

S014 · implies

Before

first map:target event index=10;event=[1, 3];mapping=[1, 2, 0, 0]

After

source membership tests=[Yes, No, No, No];preimage=[0];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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S013

S015 · implies

Before

first map:target event index=11;event=[0, 1, 3];mapping=[1, 2, 0, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S014

S016 · implies

Before

first map:target event index=12;event=[2, 3];mapping=[1, 2, 0, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S015

S017 · implies

Before

first map:target event index=13;event=[0, 2, 3];mapping=[1, 2, 0, 0]

After

source membership tests=[No, Yes, Yes, Yes];preimage=[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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S016

S018 · implies

Before

first map:target event index=14;event=[1, 2, 3];mapping=[1, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S017

S019 · implies

Before

first map:target event index=15;event=[0, 1, 2, 3];mapping=[1, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S018

S020 · implies

Before

first map:all target preimage rows=[[[], [], 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]]

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 dependencies

Rule: measurability_all_target_events_conjunction · Uses: G001, S019

S021 · implies

Before

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 dependencies

Rule: qualify_complete_finite_sigma_algebra · Uses: G001, S020

S022 · implies

Before

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 dependencies

Rule: qualify_complete_finite_sigma_algebra · Uses: G001, S021

S023 · implies

Before

second map:mapping=[2, 3, 2, 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 dependencies

Rule: measurable_total_map_and_direction · Uses: G001, S022

S024 · implies

Before

second map:target event index=0;event=[];mapping=[2, 3, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S023

S025 · implies

Before

second map:target event index=1;event=[0];mapping=[2, 3, 2, 0]

After

source membership tests=[No, No, No, Yes];preimage=[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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S024

S026 · implies

Before

second map:target event index=2;event=[1];mapping=[2, 3, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S025

S027 · implies

Before

second map:target event index=3;event=[0, 1];mapping=[2, 3, 2, 0]

After

source membership tests=[No, No, No, Yes];preimage=[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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S026

S028 · implies

Before

second map:target event index=4;event=[2];mapping=[2, 3, 2, 0]

After

source membership tests=[Yes, No, Yes, No];preimage=[0, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S027

S029 · implies

Before

second map:target event index=5;event=[0, 2];mapping=[2, 3, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S028

S030 · implies

Before

second map:target event index=6;event=[1, 2];mapping=[2, 3, 2, 0]

After

source membership tests=[Yes, No, Yes, No];preimage=[0, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S029

S031 · implies

Before

second map:target event index=7;event=[0, 1, 2];mapping=[2, 3, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S030

S032 · implies

Before

second map:target event index=8;event=[3];mapping=[2, 3, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S031

S033 · implies

Before

second map:target event index=9;event=[0, 3];mapping=[2, 3, 2, 0]

After

source membership tests=[No, Yes, No, Yes];preimage=[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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S032

S034 · implies

Before

second map:target event index=10;event=[1, 3];mapping=[2, 3, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S033

S035 · implies

Before

second map:target event index=11;event=[0, 1, 3];mapping=[2, 3, 2, 0]

After

source membership tests=[No, Yes, No, Yes];preimage=[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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S034

S036 · implies

Before

second map:target event index=12;event=[2, 3];mapping=[2, 3, 2, 0]

After

source membership tests=[Yes, Yes, Yes, No];preimage=[0, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S035

S037 · implies

Before

second map:target event index=13;event=[0, 2, 3];mapping=[2, 3, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S036

S038 · implies

Before

second map:target event index=14;event=[1, 2, 3];mapping=[2, 3, 2, 0]

After

source membership tests=[Yes, Yes, Yes, No];preimage=[0, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S037

S039 · implies

Before

second map:target event index=15;event=[0, 1, 2, 3];mapping=[2, 3, 2, 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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S038

S040 · implies

Before

second map:all target preimage rows=[[[], [], Yes], [[0], [3], Yes], [[1], [], Yes], [[0, 1], [3], Yes], [[2], [0, 2], Yes], [[0, 2], [0, 2, 3], Yes], [[1, 2], [0, 2], Yes], [[0, 1, 2], [0, 2, 3], Yes], [[3], [1], Yes], [[0, 3], [1, 3], Yes], [[1, 3], [1], Yes], [[0, 1, 3], [1, 3], Yes], [[2, 3], [0, 1, 2], Yes], [[0, 2, 3], [0, 1, 2, 3], Yes], [[1, 2, 3], [0, 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 dependencies

Rule: measurability_all_target_events_conjunction · Uses: G001, S039

S041 · implies

Before

source label=0;first image=1;second table=[2, 3, 2, 0]

After

second-after-first image=3

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 dependencies

Rule: composition_declared_order_lookup · Uses: G001, S040

S042 · implies

Before

source label=1;first image=2;second table=[2, 3, 2, 0]

After

second-after-first image=2

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 dependencies

Rule: composition_declared_order_lookup · Uses: G001, S041

S043 · implies

Before

source label=2;first image=0;second table=[2, 3, 2, 0]

After

second-after-first image=2

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 dependencies

Rule: composition_declared_order_lookup · Uses: G001, S042

S044 · implies

Before

source label=3;first image=0;second table=[2, 3, 2, 0]

After

second-after-first image=2

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 dependencies

Rule: composition_declared_order_lookup · Uses: G001, S043

S045 · implies

Before

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 dependencies

Rule: qualify_complete_finite_sigma_algebra · Uses: G001, S044

S046 · implies

Before

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 dependencies

Rule: qualify_complete_finite_sigma_algebra · Uses: G001, S045

S047 · implies

Before

composite map:mapping=[3, 2, 2, 2];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 dependencies

Rule: measurable_total_map_and_direction · Uses: G001, S046

S048 · implies

Before

composite map:target event index=0;event=[];mapping=[3, 2, 2, 2]

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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S047

S049 · implies

Before

composite map:target event index=1;event=[0];mapping=[3, 2, 2, 2]

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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S048

S050 · implies

Before

composite map:target event index=2;event=[1];mapping=[3, 2, 2, 2]

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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S049

S051 · implies

Before

composite map:target event index=3;event=[0, 1];mapping=[3, 2, 2, 2]

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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S050

S052 · implies

Before

composite map:target event index=4;event=[2];mapping=[3, 2, 2, 2]

After

source membership tests=[No, Yes, Yes, Yes];preimage=[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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S051

S053 · implies

Before

composite map:target event index=5;event=[0, 2];mapping=[3, 2, 2, 2]

After

source membership tests=[No, Yes, Yes, Yes];preimage=[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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S052

S054 · implies

Before

composite map:target event index=6;event=[1, 2];mapping=[3, 2, 2, 2]

After

source membership tests=[No, Yes, Yes, Yes];preimage=[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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S053

S055 · implies

Before

composite map:target event index=7;event=[0, 1, 2];mapping=[3, 2, 2, 2]

After

source membership tests=[No, Yes, Yes, Yes];preimage=[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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S054

S056 · implies

Before

composite map:target event index=8;event=[3];mapping=[3, 2, 2, 2]

After

source membership tests=[Yes, No, No, No];preimage=[0];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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S055

S057 · implies

Before

composite map:target event index=9;event=[0, 3];mapping=[3, 2, 2, 2]

After

source membership tests=[Yes, No, No, No];preimage=[0];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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S056

S058 · implies

Before

composite map:target event index=10;event=[1, 3];mapping=[3, 2, 2, 2]

After

source membership tests=[Yes, No, No, No];preimage=[0];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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S057

S059 · implies

Before

composite map:target event index=11;event=[0, 1, 3];mapping=[3, 2, 2, 2]

After

source membership tests=[Yes, No, No, No];preimage=[0];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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S058

S060 · implies

Before

composite map:target event index=12;event=[2, 3];mapping=[3, 2, 2, 2]

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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S059

S061 · implies

Before

composite map:target event index=13;event=[0, 2, 3];mapping=[3, 2, 2, 2]

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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S060

S062 · implies

Before

composite map:target event index=14;event=[1, 2, 3];mapping=[3, 2, 2, 2]

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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S061

S063 · implies

Before

composite map:target event index=15;event=[0, 1, 2, 3];mapping=[3, 2, 2, 2]

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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S062

S064 · implies

Before

composite map:all target preimage rows=[[[], [], 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]]

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 dependencies

Rule: measurability_all_target_events_conjunction · Uses: G001, S063

S065 · implies

Before

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 dependencies

Rule: 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, [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)

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: 22f6b9d0a0ca193a7a32b338967c221e34753f14b85690c70efbf37b42e1562f
  • candidate record SHA-256: 5dc39c3c6aa6b970f15990e8e5da15ebc25fede1247f634548a3b127d9ad7c6e
  • semantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258
  • question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON