Conditional expectation ($\mathbb{E}[X\mid \mathcal{G}]$)

For an integrable random variable $X$ and a sub-sigma-algebra $\mathcal{G}$, the conditional expectation $\mathbb{E}[X\mid \mathcal{G}]$ is the almost-everywhere unique $\mathcal{G}$-measurable integrable random variable satisfyingfor every $G\in \mathcal{G}$.