Existence and Uniqueness of Conditional Expectation for Square-Integrable Random Variables

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$, and let $X$ be a square-integrable random variable on it. Call a random variable $Z$ $\mathcal{G}$-measurable if $Z^{-1}(B)\in\mathcal{G}$ for every Borel set $B$, and write $\mathbf{1}_{A}$ for the function equal to $1$ on $A$ and $0$ off $A$.

Then there exists a $\mathcal{G}$-measurable square-integrable random variable $Y$ with the following three properties, where norms, inner products, and expectations are those of Square-Integrable Random Variables and the Mean-Square Inner Product:

1. (Best approximation) $\lVert X-Y\rVert_{2}\le\lVert X-Z\rVert_{2}$ for every $\mathcal{G}$-measurable square-integrable random variable $Z$;

2. (Orthogonality) $\mathbb{E}\bigl[(X-Y)Z\bigr]=0$ for every $\mathcal{G}$-measurable square-integrable random variable $Z$;

3. (Averaging property) $\mathbb{E}[X\mathbf{1}_{A}]=\mathbb{E}[Y\mathbf{1}_{A}]$ for every $A\in\mathcal{G}$.

Moreover, for a $\mathcal{G}$-measurable square-integrable random variable $Y'$, the following are equivalent: $Y'$ satisfies property 1; $Y'$ satisfies property 2; $Y'$ satisfies property 3. Finally, uniqueness holds up to almost-sure equality in the following precise sense: if $Y$ and $Y'$ are $\mathcal{G}$-measurable square-integrable random variables that each satisfy property 3 (equivalently, property 1 or 2), then $P(Y=Y')=1$.