Conditional Expectation

Let $X \in L^1(\Omega, \mathcal{F}, \mathbb{P})$ (Definition~Expectation) and let $\mathcal{G} \subseteq \mathcal{F}$ be a sub-$\sigma$-algebra (Definition~Sigma-Algebra). The conditional expectation $\mathbb{E}[X \mid \mathcal{G}]$ is the $\mathbb{P}$-a.s.\ unique $\mathcal{G}$-measurable random variable satisfying \[ \int_G \mathbb{E}[X \mid \mathcal{G}]\, d\mathbb{P} = \int_G X\, d\mathbb{P} \quad for all G \in \mathcal{G}. \] Its existence and uniqueness follow from the Radon--Nikod\'{y}m theorem.