Conditional expectation

Remarks

The existence follows from the Radon–Nikodym theorem. For $E[X^2] < \infty$, $E[X \mid \mathcal{G}]$ is the orthogonal projection of $X$ onto $L^2(\mathcal{G})$. For a discrete $Y$, $E[X \mid Y] = g(Y)$ with $g(y) = E[X \, 1_{\{Y = y\}}] / P(Y = y)$ when $P(Y = y) > 0$. This $g(y)$ is the expectation of $X$ under the conditional probability $P(\cdot \mid Y = y)$. The conditional expectation is linear and monotone. If $W$ is $\mathcal{G}$-measurable and $WX$ is integrable, then $E[WX \mid \mathcal{G}] = W \, E[X \mid \mathcal{G}]$. If $X$ is independent of $\mathcal{G}$, then $E[X \mid \mathcal{G}] = E[X]$.