Conditional Expectation with respect to Random Element

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space, let $X$ be a Random Variable in $L^{1}(\mathcal{B})$, and let $Y: (\Omega, \mathcal{B}) \to (S, \mathcal{S})$ bea Random Element. Then the Conditional Expectation of $X$ with respect to $Y$ is the (Almost Surely) Random Variable $\mathbb{E}[X | \sigma(Y)]$.

• Sigma Algebra induced by Random Element