LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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Conditioning a known state and independent noise

Statement

Assume the Axiom of Choice for the conditional-expectation interface. Let $(Ω, F, P)$ be a probability space, let $G \subseteq F$ be a sub-sigma-algebra, let $X$ be an $(S, Σ)$ -valued random element that is $G$ -measurable, and let $Y$ be a $(T, T)$ -valued random element whose sigma-algebra $σ (Y)$ is independent of $G$ . Let $μ$ be the law of $Y$ on $(T, T)$ . If $h : S \times T \to R$ is bounded and $Σ \otimes T$ -measurable, then $H (x) : = \int_{T} h (x, y) μ (d y)$ is $Σ$ -measurable and $E [h (X, Y) ∣ G] = H (X) almost surely .$

Facts & Assumptions

Given: AC, a probability space, a sub-sigma-algebra $G$ , a $G$ -measurable random element $X$ , a random element $Y$ with $σ (Y)$ independent of $G$ , and a bounded product-measurable $h$ .

[F1]

Measurability of $x \mapsto \int_{T} h (x, y) K (x, d y)$ for a finite kernel $K$ , in particular a probability kernel, is theorem-level; the constant map $K (s, A) : = μ (A)$ is a probability kernel because $s \mapsto μ (A)$ is constant. Measure kernel and probability kernel Measurability of integration against a kernel

[F2]

A conditional-expectation version is characterized by its $G$ -event integrals, and versions are unique almost surely. Conditional expectation given a sigma algebra Conditional expectation as an ae class Conditional expectation is unique almost surely

[F3]

Bounded $G$ -measurable factors come out of the conditional expectation, and conditional expectation is linear on integrable inputs. Taking out what is known Basic algebra and order properties of conditional expectation

[F4]

If a random variable has the independence rectangle identity against $G$ , its conditional expectation given $G$ is its mean; this applies to $1_{C} (Y)$ for $C \in T$ because $σ (Y)$ is independent of $G$ . Conditioning a known variable and an independent variable Independent sigma-algebras and independent events Independent random elements

[F5]

The product sigma-algebra is generated by the measurable rectangles, which form a pi-system containing the whole space; a lambda-system containing a pi-system contains the generated sigma-algebra. Dynkin's pi-lambda theorem The product sigma-algebra and its finite iterates

[F6]

Sections of product-measurable sets are measurable, and compositions of a measurable map with a measurable function are measurable. Every section of a product-measurable function is measurable Closure properties of measurable functions used by the integral Law or distribution of a random element

[F7]

Nonnegative measurable functions are increasing limits of nonnegative simple functions, and monotone convergence passes those limits through integrals. Every nonnegative measurable function is the increasing limit of simple measurable functions Monotone convergence for the integral

[F8]

AC supplies the conditional-expectation existence used in [F2]. The Axiom of Choice

Proof

technique · direct

1.1

Fix a measurable rectangle $A \times C$ with $A \in Σ$ and $C \in T$ . The indicator $1_{A \times C} (X, Y) = 1_{A} (X) 1_{C} (Y)$ has bounded $G$ -measurable factor $1_{A} (X)$ , so [F3] and then [F4] give $E [1_{A \times C} (X, Y) ∣ G] = 1_{A} (X) E [1_{C} (Y) ∣ G] = 1_{A} (X) P (Y \in C)$ almost surely; since $H_{A \times C} (x) = \int_{T} 1_{A \times C} (x, y) μ (d y) = μ (C) 1_{A} (x)$ , this is the asserted identity for rectangles.

F3F4given

2.1

Let $D$ be the class of $B \in Σ \otimes T$ with $E [1_{B} (X, Y) ∣ G] = H_{B} (X)$ almost surely, where $H_{B} (x) : = μ (B_{x})$ and $B_{x} = {y \in T : (x, y) \in B}$ . Each $B_{x}$ is measurable by [F6], the constant kernel $K (s, \cdot) : = μ$ is a probability kernel and [F1] makes $H_{B}$ $Σ$ -measurable, while $H_{B} (X)$ is then $G$ -measurable and bounded by [F6]. Step 1.1 shows that $D$ contains every measurable rectangle.

F1F6step 1.1

3.1

The class $D$ is a lambda-system on $S \times T$ . It contains $S \times T$ because $H_{S \times T} (x) = μ (T) = 1$ for every $x \in S$ , so both sides of its defining conditional-expectation identity are $1$ . If $B_{1} \subseteq B_{2}$ lie in $D$ , then for every $G$ -event $G_{0}$ , subtraction of their defining event-integral identities gives $\int_{G_{0}} 1_{B_{2} ∖ B_{1}} (X, Y) d P = \int_{G_{0}} (H_{B_{2}} - H_{B_{1}}) (X) d P$ , while sectionwise $H_{B_{2} ∖ B_{1}} = H_{B_{2}} - H_{B_{1}}$ ; hence $B_{2} ∖ B_{1} \in D$ by [F2]. If $B_{1} \subseteq B_{2} \subseteq \dots$ are in $D$ with union $B$ , then for each $x$ the numbers $H_{B_{n}} (x) = μ ((B_{n})_{x})$ increase to $μ (B_{x}) = H_{B} (x)$ , and [F7] applied to the finite measure $P$ restricted to $G_{0}$ gives $\int_{G_{0}} 1_{B_{n}} (X, Y) d P = \int_{G_{0}} H_{B_{n}} (X) d P \to \int_{G_{0}} H_{B} (X) d P$ for every $G$ -event $G_{0}$ ; hence $B \in D$ by [F2].

F2F7step 2.1

4.1

The measurable rectangles form a pi-system containing the product-space whole set $S \times T$ and generate $Σ \otimes T$ , so [F5] applied to the lambda-system $D$ gives $D = Σ \otimes T$ ; that is, $E [1_{B} (X, Y) ∣ G] = H_{B} (X)$ almost surely for every product-measurable $B$ .

F5step 2.1step 3.1

5.1

Let $h \geq 0$ be bounded. By [F7] there are nonnegative simple functions $s_{n} = \sum_{j} c_{j, n} 1_{B_{j, n}}$ with $s_{n} ↑ h$ pointwise, the sets $B_{j, n}$ being product measurable and the sums finite; step 4.1 and linearity of the integral give, for every $G$ -event $G_{0}$ , $\int_{G_{0}} s_{n} (X, Y) d P = \sum_{j} c_{j, n} \int_{G_{0}} 1_{B_{j, n}} (X, Y) d P = \sum_{j} c_{j, n} \int_{G_{0}} H_{B_{j, n}} (X) d P = \int_{G_{0}} H_{s_{n}} (X) d P$ , where $H_{s_{n}} (x) = \int_{T} s_{n} (x, y) μ (d y)$ .

F7step 4.1

6.1

For bounded nonnegative $h$ , monotone convergence [F7] applied to $s_{n} (X, Y) ↑ h (X, Y)$ and to $H_{s_{n}} (X) ↑ H_{h} (X)$ in step 5.1 gives $\int_{G_{0}} h (X, Y) d P = \int_{G_{0}} H_{h} (X) d P$ for every $G$ -event $G_{0}$ . Since $H_{h}$ is $Σ$ -measurable by [F1] and bounded by $∥ h ∥_{\infty}$ , [F2] identifies $H_{h} (X)$ with $E [h (X, Y) ∣ G]$ almost surely.

F1F2F7step 5.1

7.1

For a general bounded real $h$ , apply step 6.1 to the bounded nonnegative functions $h^{+}$ and $h^{-}$ and subtract the two almost-sure identities using linearity [F3]; since $H_{h} = H_{h^{+}} - H_{h^{-}}$ pointwise, $H_{h}$ is $Σ$ -measurable and $E [h (X, Y) ∣ G] = H_{h} (X)$ almost surely. This is the asserted identity, with $H = H_{h}$ as displayed in the statement.

F1F3step 6.1

8.1

The boundary cases behave as stated and need no separate treatment: $h \equiv 0$ gives $H \equiv 0$ and both sides vanish; if $T$ is a single point with its unique probability measure, then $H (x) = h (x, y_{0})$ and the statement reduces to $E [h (X, y_{0}) ∣ G] = h (X, y_{0})$ for the $G$ -measurable $X$ , which is [F3] with a deterministic factor; if $A = \emptyset$ or $C = \emptyset$ the rectangle identity of step 1.1 reads $0 = 0$ . The independence hypothesis is used exactly once, in step 1.1, and no other selection is made; AC is used only through [F8] in [F2].

F3F8givenstep 1.1∎

Source notes

Durrett's proof of Theorem 7.2.1 conditions on the known value of $B_{s}$ and on the independent increment; van der Vaart, Section 1.4, records the rectangle-to-product-sigma-algebra Dynkin extension in this generality. The proof above isolates that extension as a lemma because the Brownian Markov, future-path and planar arguments all consume it with different state spaces $S$ and $T$ .

Depends on

Used by

Planar Brownian annular exit probability Lemma
Brownian reflection principle Theorem
Future-path Markov property Theorem
Markov property of Brownian motion Theorem

Dependency tree · two levels

56 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

Rick Durrett, Probability: Theory and Examples, fifth edition, proof of Theorem 7.2.1 (standard reference, not scraped)
Aad van der Vaart, Martingales, Diffusions and Financial Mathematics, Section 1.4 (standard reference, not scraped)