Mean-Square Completeness of Square-Integrable Random Variables (Riesz-Fischer)

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$, and let $(X_n)_{n\in\mathbb{N}}$ be a sequence of square-integrable random variables on $(\Omega,\mathcal{F},P)$, each $\mathcal{G}$-measurable in the sense that $X_n^{-1}(B)\in\mathcal{G}$ for every Borel set $B$. Assume the sequence is Cauchy in mean square: for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ such that the mean-square distance of Square-Integrable Random Variables and the Mean-Square Inner Product satisfies

Then there exists a $\mathcal{G}$-measurable square-integrable random variable $X$ on $(\Omega,\mathcal{F},P)$ such that the real sequence $(\lVert X_n-X\rVert_{2})_{n\in\mathbb{N}}$ has limit $0$.

Moreover, $X$ is unique up to almost-sure equality in the following precise sense: if $X'$ is any square-integrable random variable on $(\Omega,\mathcal{F},P)$ with $\lVert X_n-X'\rVert_{2}\to0$, then $P(X=X')=1$.

In particular, taking $\mathcal{G}=\mathcal{F}$: every mean-square Cauchy sequence of square-integrable random variables converges in mean square to a square-integrable random variable.