Measurable function

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

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 — Fix the measurable set families of both source and target first.
• Function mapping — A complete total map determines preimages.
• 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.