Conditional Expectation with respect to Sigma Field

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space and let $X$ be a Random Variable in $L^{1}(\mathcal{B})$. Let $\mathcal{G} \subset \mathcal{B}$ be a Sub-Sigma Algebra. Then the Conditional Expectation of $X$ (denoted $\mathbb{E}[X | \mathcal{G}]$) with respect to $\mathcal{G}$ is a Random Variable such that

1. $\mathbb{E}[X|\mathcal{G}] \in \mathcal{G}$
2. $\forall A \in \mathcal{G}$, $$\int\limits_{A} \mathbb{E}[X|\mathcal{G}] d \mathbb{P}(\omega) = \int\limits_{A} X d \mathbb{P}(\omega).$$

1. Conditional Expectation Exists and is Almost Surely Unique
2. TODO