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Integrable factorization under conditional independence

Statement

Assume the The Axiom of Choice for the conditional-expectation classes. Let $U$ be a real integrable random variable, let $Z$ be a random element, and let $G$ be a sub-sigma-algebra. If $U$ and $Z$ are conditionally independent given $G$ in the bounded-function sense of Conditional independence given a sigma-algebra, then for every bounded measurable real function $h$ on the state space of $Z$ ,

$E [U h (Z) ∣ G] = E [U ∣ G] E [h (Z) ∣ G] almost surely .$

Facts & Assumptions

Given: The Axiom of Choice, $U \in L^{1} (P)$ , a random element $Z$ , a sub-sigma-algebra $G$ , and $U ⊥ ⁣ ⁣ ⁣ ⊥ Z ∣ G$ .

[A1]

The Axiom of Choice is assumed for the conditional-expectation classes used in this lemma. The Axiom of Choice

[F1]

Conditional independence is the bounded-product conditional-expectation identity. Conditional independence given a sigma-algebra

[F2]

Conditional-expectation notation denotes the almost-sure class of versions, which exists under the stated Choice assumption. Conditional expectation as an ae class

[F3]

Under Choice, conditional expectation contracts every $L^{p}$ norm for $1 \leq p \leq \infty$ , in particular $L^{1}$ and $L^{\infty}$ . Conditional lp contraction

Proof

1.1givenF1F2construct

For each integer $n \geq 1$ , let $T_{n} (u) = \max (- n, \min (u, n))$ . The functions $T_{n} (U)$ and $h (Z)$ are bounded and measurable, so [F1] gives the identity $E [T_{n} (U) h (Z) ∣ G] = E [T_{n} (U) ∣ G] E [h (Z) ∣ G]$ of the classes in [F2].

2.1A1F2F3step 1.1algebra∎

Since $U \in L^{1}$ , $T_{n} (U) \to U$ in $L^{1}$ ; boundedness of $h$ gives $T_{n} (U) h (Z) \to U h (Z)$ in $L^{1}$ . By [F3], the conditional expectations on the left converge in $L^{1}$ to $E [U h (Z) ∣ G]$ , and $E [T_{n} (U) ∣ G] \to E [U ∣ G]$ in $L^{1}$ . Also $∣ E [h (Z) ∣ G] ∣ \leq ∥ h ∥_{\infty}$ almost surely by [F3], so multiplication by this factor preserves that $L^{1}$ convergence. Taking the $L^{1}$ limits in step 1.1 proves the stated identity of almost-sure classes.

Depends on

Used by

Discrete-treatment potential-outcome means are identifiable Theorem

Dependency tree · two levels

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Sources

David Aldous and Sinho Chewi, Probability Theory lecture notes, Lecture 9 (standard reference, not scraped)