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 $\mathcal G \subseteq \Sigma$ be a sub-$\sigma$-algebra of $\Sigma$.

We say that $Z$ is a version of the conditional expectation of $X$ given $\mathcal G$, or version of $\expect {X \mid \mathcal G}$ :


 * $(1): \quad$ $\expect {\cmod Z} < \infty$
 * $(2): \quad$ $Z$ is $\mathcal G$-measurable
 * $(3): \quad$ We have:
 * $\ds \int_G Z \rd \Pr = \int_G X \rd \Pr$ for each $G \in \mathcal G$.

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


 * $Z = \expect {X \mid \mathcal G}$

in the sense of almost-sure equality.