Definition:Conditional Expectation/General Case/Sigma-Algebra

Definition

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

Let $X$ be an integrable random variable on $\struct {\Omega, \Sigma, \Pr}$.

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

We say that $Z$ is a version of the conditional expectation of $X$ given $\GG$, or version of $\expect {X \mid \GG}$ if and only if:

$(1): \quad \expect {\cmod Z} < \infty$

$(2): \quad$ $Z$ is $\GG$-measurable

$(3): \quad \ds \forall G \in \GG: \int_G Z \rd \Pr = \int_G X \rd \Pr$

From Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra, any two versions of the conditional expectation of $X$ given $\GG$ agree almost surely, so we write:

$Z = \expect {X \mid \GG}$

in the sense of almost-sure equality.

Sources

• 1991: David Williams: Probability with Martingales ... (previous) ... (next): $9.2$