# Finite conditional expectation

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

Relative to a partition-generated sigma-algebra on a finite probability space, conditional expectation takes the weighted mean on each positive-mass block and preserves block integrals.

mko-conditional-expectation · definition · v0.10.0



## Mathematical statement

On the same finite probability space, E[X|G] must be G-measurable and have the same integral as X on every G-measurable event. When G is generated by a partition, its value is the weighted mean on each positive-mass block. On zero-mass blocks it is determined only almost surely; division by zero is not performed.





## Representative formula (exact TeX)

E(X,G)





## Assumptions


- The complete finite outcomes, exact rational masses, X values and conditioning partition belong to the same probability space.
- The main families use positive-mass blocks; the zero-mass boundary records almost-sure nonuniqueness without forcing point values.





## Symbol scope


- E — conditional expectation
- X — finite random variable
- G — conditioning information





## Intuition, definition, and misconceptions

The partition blocks generate a finite sigma-algebra.

On each positive-mass block, conditional expectation is the mass-weighted mean.

The result is constant on each block and preserves its integral; null-block values are determined only almost surely.

Pointwise conditioning, dividing on a zero-mass block, or checking only total expectation does not establish the complete conditional-expectation obligations.





## Adopted derivation guide

[Finite conditional expectation: computing E[X|G] for a partition-generated σ-algebra, including a zero-mass block](https://ocme.evemisslab.com/en/derivations/derivation-mko-conditional-expectation-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Finite joint distribution](https://ocme.evemisslab.com/en/mko/mko-finite-joint-distribution/) — The common finite law provides exact masses.
- [Measurable function](https://ocme.evemisslab.com/en/mko/mko-measurable-function/) — The result must be measurable relative to G.
- [Expectation in finite models](https://ocme.evemisslab.com/en/mko/mko-expectation/) — Block integrals and total expectation use the same operation.
- [Conditional probability in finite models](https://ocme.evemisslab.com/en/mko/mko-conditional-probability/) — Weighted means on positive-mass blocks use the conditional law.
- [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-random-variable/) — X is defined on the same probability space.





## Verification boundary

Only complete finite partition models are handled; null-block point values are not required to agree, and no general measure-theoretic extension is inferred.

not_formalized

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





## Next obligation

Together with filtrations, add martingales.





## 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/conditional-expectation.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-conditional-expectation/index.md)