Conditional expectation

A conditional expectation of a random variable $X$ given a sub-sigma-algebra $\mathcal G \subseteq \mathcal F$ on a probability space $(\Omega,\mathcal F,\mathbb P)$ is any $\mathcal G$-measurable function $Y$ such that $X$ is integrable (i.e. $\mathbb E[|X|]<\infty$) and

Any two versions of $Y$ that satisfy this are equal almost surely, and one writes $Y=\mathbb E[X\mid \mathcal G]$.

Conditional expectation refines expectation by restricting to information contained in $\mathcal G$; the special case $X=\mathbf 1_A$ yields conditional probability of an event $A$ given $\mathcal G$.

Examples:

• If $\mathcal G=\{\varnothing,\Omega\}$ is the trivial $\sigma$-algebra, then $\mathbb E[X\mid\mathcal G]=\mathbb E[X]$ (a constant random variable).
• If $X$ is $\mathcal G$-measurable (in particular if $\mathcal G=\mathcal F$), then $\mathbb E[X\mid\mathcal G]=X$ almost surely.
• If $B\in\mathcal F$ with $\mathbb P(B)\in(0,1)$ and $\mathcal G=\sigma(B)=\{\varnothing,B,B^c,\Omega\}$, then $\mathbb E[X\mid \mathcal G] \;=\; \frac{\mathbb E[X\,\mathbf 1_B]}{\mathbb P(B)}\,\mathbf 1_B \;+\; \frac{\mathbb E[X\,\mathbf 1_{B^c}]}{\mathbb P(B^c)}\,\mathbf 1_{B^c}.$