Conditional Expectation is Linear

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{B}, \mathbb{P})$ be a Probability Space and let $X, Y$ be Random Variables in $\bar{L^{1}}(\mathcal{B})$. Let $\mathcal{G} \subset \mathcal{B}$ be a sub-Sigma Algebra of $\mathcal{B}$. Then,

$$\mathbb{E}(cX + Y | \mathcal{G}) = c \mathbb{E}(X|\mathcal{G}) + \mathbb{E}(Y|\mathcal{G})$$

TODO