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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Tower property of conditional expectation

Statement

Assume AC for existence. If $H \subseteq G \subseteq F$ and $X \in L^{1} (P)$ , then $E [E [X ∣ G] ∣ H] = E [X ∣ H]$ and $E [E [X ∣ H] ∣ G] = E [X ∣ H]$ almost surely. Also, if a version of $E [X ∣ G]$ is $H$ -measurable, it is a version of $E [X ∣ H]$ .

Facts & Assumptions

Given: AC, real integrable X, and $H \subseteq G \subseteq F$ ; for the last clause a G-conditional version is H-measurable.

[F1]

Conditional versions are measurable and integrable and have the defining event integrals. (Conditional expectation as an ae class)

[F2]

Versions for the same input and sigma-algebra agree almost surely. (Conditional expectation is unique almost surely)

Proof

technique · direct

1.1

Set $U = E [X ∣ G]$ and take a version $V = E [U ∣ H]$ . For $A \in H \subseteq G$ , $\int_{A} V = \int_{A} U = \int_{A} X$ . Since $V$ is $H$ -measurable and integrable, [F2] identifies it with $E [X ∣ H]$ .

F1F2

2.1

A version $W = E [X ∣ H]$ is already $G$ -measurable and integrable. It has its own event integrals $\int_{A} W = \int_{A} W$ on every $A \in G$ , so it is a version of $E [W ∣ G]$ ; [F2] gives the second identity. Finally if $U$ itself is $H$ -measurable, its $G$ event identities restrict to $H$ , so [F2] gives the additional assertion.

F1F2step 1.1∎

Source notes

Durrett Theorems 4.1.12–4.1.13, printed p.212; van der Vaart Lemma 1.9(v), printed p.4.

Depends on

Used by

Dependency tree · two levels

11 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)