# Measurable function

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

With fixed source and target sigma-algebras, a function is measurable when the preimage of every target measurable set belongs to the source sigma-algebra.

mko-measurable-function · definition · v0.10.0



## Mathematical statement

Given finite measurable spaces (X,Sigma_X) and (Y,Sigma_Y), a total function f:X→Y is measurable if f⁻¹(B) belongs to Sigma_X for every B in Sigma_Y. This slice checks each preimage by complete finite enumeration.





## Representative formula (exact TeX)

M(f)





## Assumptions


- Carriers use finite labels starting at 0, sigma-algebras are complete canonical subset families and functions are complete total-map tables.
- Complete finite enumeration establishes only the specified instance. Measurability does not imply continuity, integrability or measure preservation.





## Symbol scope


- M — measurability record
- f — total map





## Intuition, definition, and misconceptions

The obligation goes from target measurable sets to their inverse images.

Measurability of an indicator is controlled by its event and the event’s complement in the source sigma-algebra.

Composition of two measurable maps can be read back through composition of preimages.

Do not check forward images instead of preimages or identify measurability with continuity, integrability or measure preservation.





## Adopted derivation guide

[Measurable function: checking every preimage by complete enumeration, with composition, indicators and the image-set trap](https://ocme.evemisslab.com/en/derivations/derivation-mko-measurable-function-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/mko/mko-sigma-algebra-measure-space/) — Fix the measurable set families of both source and target first.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — A complete total map determines preimages.
- [Set operations](https://ocme.evemisslab.com/en/mko/mko-set-operations/) — Preimages, complements and unions use set operations.





## Verification boundary

Complete finite enumeration does not extend to arbitrary infinite measurable spaces.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add simple functions, integration and measure convergence.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/measurable-function.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-measurable-function/index.md)