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

1.1F1F2F3

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 $G$ .

2.1step 1.1F4∎

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.

Source notes

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

Depends on

Used by

One-dimensional Brownian motion is recurrent Corollary
A submartingale need not have increasing sample paths Counterexample
An unbounded predictable transform may lose integrability Counterexample
Marginal sub-Gaussianity does not justify an anytime confidence sequence Counterexample
Strong Markov fails at a nonstopping random time Counterexample
Conditional expectation as an ae class Definition
Conditional law given a random element Definition
A predictable linear forecast and its empirical residual process Example
Conditional expectation given a discrete random variable Example
Conditioning on a finite partition Example
Dyadic conditional expectation martingale Example
Polya urn proportion martingale Example
A finite label has a conditional probability function Lemma
Bounded-function form of the Markov property Lemma
Conditional-independence equivalences and preservation Lemma
Conditioning a known state and independent noise Lemma
Conditioning a known variable and an independent variable Lemma
Elementary Ito integrals do not depend on step representation Lemma
Planar Brownian annular exit probability Lemma
The posterior mean is the unique squared-loss Bayes action in a Hilbert target Proposition
A Gaussian finite-history forecast is a conditional expectation Theorem
Basic algebra and order properties of conditional expectation Theorem
Blumenthal's zero-one law Theorem
Conditional integration through a regular conditional law Theorem
Conditional monotone convergence Theorem
Conditional-class standardization identifies a causal mean Theorem
Discrete-treatment potential-outcome means are identifiable Theorem
Future-path Markov property Theorem
Gaussian simple kriging is a conditional mean Theorem
Inverse probability weighting identifies a causal mean Theorem
Ito isometry for elementary integrands Theorem
Markov property of Brownian motion Theorem
Strong Markov property of Brownian motion Theorem
Taking out what is known Theorem
Terminal error bounds for a sequential likelihood-ratio test Theorem
Tower property of conditional expectation Theorem
Two-sided Brownian exit probability Theorem

Dependency tree · two levels

18 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)