Conditional Expectation of Measurable Random Variable

Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.

Let $X$ be a integrable random variable that is $\GG$-measurable.

Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.

Then:

$\expect {X \mid \GG} = X$ almost everywhere.

Proof

We show that $X$ is a version of $\expect {X \mid \GG}$.

Then since conditional expectation is essentially unique by Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra, we will obtain:

$\expect {X \mid \GG} = X$ almost everywhere.

Note that $X$ is integrable and $\GG$-measurable by hypothesis, with:

$\ds \int_A X \rd \Pr = \int_A X \rd \Pr$

So $X$ satisfies the conditions of conditional expectation, and so:

$\expect {X \mid \GG} = X$ almost everywhere.

$\blacksquare$

Sources

• 1991: David Williams: Probability with Martingales ... (previous) ... (next): $9.7$: Properties of conditional expectation: a list