LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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Conditional expectation is unique almost surely

Statement

If $Y, Z$ are conditional-expectation versions of the same real integrable $X$ given $G$ , then $Y = Z$ almost surely.

Facts & Assumptions

Given: A probability space, a sub-sigma-algebra $G$ , an integrable real X, and two versions Y,Z with all its G-event integrals.

[F1]

Both versions have the same event integrals. (Conditional expectation given a sigma algebra)

[F2]

The difference is integrable and its integral is the difference of integrals. (The Lebesgue integral is linear on $L^{1} (μ)$ )

[F3]

Differences and their positive and negative parts are measurable. (Closure properties of measurable functions used by the integral)

[F4]

Zero integral of a nonnegative function implies it vanishes almost everywhere. (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere)

Proof

technique · direct

1.1

The difference $D = Y - Z$ is $G$ -measurable and integrable, and $\int_{A} D d P = 0$ for every $A \in G$ . In particular the sets $A_{+} = {D > 0}$ and $A_{-} = {D < 0}$ belong to $G$ .

F1F2F3

2.1

On $A_{+}$ , $D 1_{A_{+}} = D^{+} \geq 0$ has zero integral; on $A_{-}$ , $- D 1_{A_{-}} = D^{-} \geq 0$ also has zero integral. By [F4], both parts vanish almost surely. Off the union of their two null exceptional sets, $D = D^{+} - D^{-} = 0$ , proving $Y = Z$ almost surely.

step 1.1F4∎

Source notes

Durrett §4.1, uniqueness paragraph, printed p.206; van der Vaart Theorem 1.3, printed p.2.

Depends on

Used by

A submartingale need not have increasing sample paths Counterexample
An unbounded predictable transform may lose integrability Counterexample
Conditional expectation as an ae class Definition
Conditional law given a random element Definition
Conditional expectation given a discrete random variable Example
Conditioning on a finite partition Example
Dyadic conditional expectation martingale Example
Polya urn proportion martingale Example
Conditioning a known variable and an independent variable Lemma
Basic algebra and order properties of conditional expectation Theorem
Conditional integration through a regular conditional law Theorem
Conditional monotone convergence Theorem
Taking out what is known Theorem
Tower property of conditional expectation Theorem

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Durrett, Probability: Theory and Examples, 5th ed. (standard reference, not scraped)