Finite conditional expectation

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

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 — The common finite law provides exact masses.
• Measurable function — The result must be measurable relative to G.
• Expectation in finite models — Block integrals and total expectation use the same operation.
• Conditional probability in finite models — Weighted means on positive-mass blocks use the conditional law.
• Real random variables on finite sample spaces — 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.