Question 11473: Audit Every Target Measurable Set by Preimage

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0021-11473 · q-20261013-advance-20261024-r02-measurable-function-0048

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=[[], [1], [0, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 1, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

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=[[], [1], [0, 2, 3], [0, 1, 2, 3]], target size=4, target sigma=[[], [0], [1], [0, 1], [2], [0, 2], [1, 2], [0, 1, 2], [3], [0, 3], [1, 3], [0, 1, 3], [2, 3], [0, 2, 3], [1, 2, 3], [0, 1, 2, 3]], mapping=[2, 1, 1, 0]. Give (source size,target size,mapping,every [target measurable set,its preimage,preimage is source-measurable?] row,all rows pass?).

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

Step-by-step derivation

S001 · implies

Before

given map source:carrier Fin(4);complete canonical sigma family=[[], [1], [0, 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

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

Step identifiers and dependencies

Rule: qualify_complete_finite_sigma_algebra · Uses: G001, S001

S003 · implies

Before

given map:mapping=[2, 1, 1, 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

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

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

After

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

Rule: every_target_event_preimage_membership · Uses: G001, S004

S006 · implies

Before

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

After

source membership tests=[No, Yes, Yes, No];preimage=[1, 2];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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S005

S007 · implies

Before

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

Rule: every_target_event_preimage_membership · Uses: G001, S006

S008 · implies

Before

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

Rule: every_target_event_preimage_membership · Uses: G001, S007

S009 · implies

Before

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

After

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

Rule: every_target_event_preimage_membership · Uses: G001, S008

S010 · implies

Before

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

After

source membership tests=[Yes, Yes, Yes, No];preimage=[0, 1, 2];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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S009

S011 · implies

Before

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

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

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

After

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

Rule: every_target_event_preimage_membership · Uses: G001, S012

S014 · implies

Before

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

After

source membership tests=[No, Yes, Yes, No];preimage=[1, 2];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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S013

S015 · implies

Before

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

Rule: every_target_event_preimage_membership · Uses: G001, S014

S016 · implies

Before

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

Rule: every_target_event_preimage_membership · Uses: G001, S015

S017 · implies

Before

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

After

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

Rule: every_target_event_preimage_membership · Uses: G001, S016

S018 · implies

Before

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

After

source membership tests=[Yes, Yes, Yes, No];preimage=[0, 1, 2];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 dependencies

Rule: every_target_event_preimage_membership · Uses: G001, S017

S019 · implies

Before

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

given map:all target preimage rows=[[[], [], Yes], [[0], [3], No], [[1], [1, 2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]]

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 dependencies

Rule: 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, 1, 1, 0], [[[], [], Yes], [[0], [3], No], [[1], [1, 2], No], [[0, 1], [1, 2, 3], No], [[2], [0], No], [[0, 2], [0, 3], No], [[1, 2], [0, 1, 2], No], [[0, 1, 2], [0, 1, 2, 3], Yes], [[3], [], Yes], [[0, 3], [3], No], [[1, 3], [1, 2], No], [[0, 1, 3], [1, 2, 3], No], [[2, 3], [0], No], [[0, 2, 3], [0, 3], No], [[1, 2, 3], [0, 1, 2], No], [[0, 1, 2, 3], [0, 1, 2, 3], Yes]], No)

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: 5fe351da30720f3434fa94e3a4fc54dcf1e1ab137b5b28a8b3c2421b971fc4c3
candidate record SHA-256: c1ac1e0fd2af55ec03a043d6ccca568acc5fcdef9aa9caaad1873025c01be2f2
semantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258
question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139
Lean: NOT_RUN · human review: NOT_PERFORMED

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