Proof of Joint Continuity of Conditional Expectation under Mean-Square Convergence

Fix conditional expectations $Y_n$ of $X_n$ given $\mathcal{G}_n$ for each $n\in\mathbb{N}$ and $Y$ of $X$ given $\mathcal{G}$. For each $n$, also fix a conditional expectation $W_n$ of $X$ given $\mathcal{G}_n$, which exists by Existence and Uniqueness of Conditional Expectation for Square-Integrable Random Variables. All norms below are the mean-square norm.

Step 1: Comparison at the same $\sigma$-algebra. By the linearity property (property 1 of Basic Properties of Conditional Expectation for Square-Integrable Random Variables, with coefficients $1$ and $-1$), the random variable $Y_n-W_n$ is a conditional expectation of $X_n-X$ given $\mathcal{G}_n$. By the mean-square contraction property (property 6 of Basic Properties of Conditional Expectation for Square-Integrable Random Variables),

$$\lVert Y_n-W_n\rVert_{2}\le\lVert X_n-X\rVert_{2}.$$

Step 2: Comparison of the $\sigma$-algebras at the fixed variable $X$. Since $(\mathcal{G}_n)_{n\in\mathbb{N}}$ converges in mean square to $\mathcal{G}$, applying that definition to the square-integrable random variable $X$ and to the choices $W_n$ and $Y$ gives that the real sequence $(\lVert W_n-Y\rVert_{2})_{n\in\mathbb{N}}$ has limit $0$.

Step 3: Conclusion. By the triangle inequality applied to $Y_n-Y=(Y_n-W_n)+(W_n-Y)$, together with Step 1,

$$\lVert Y_n-Y\rVert_{2}\le\lVert Y_n-W_n\rVert_{2}+\lVert W_n-Y\rVert_{2}\le\lVert X_n-X\rVert_{2}+\lVert W_n-Y\rVert_{2}.$$

Given a real $\varepsilon>0$, the hypothesis $\lVert X_n-X\rVert_{2}\to0$ and Step 2 provide $N_1,N_2\in\mathbb{N}$ with $\lVert X_n-X\rVert_{2}<\varepsilon/2$ for $n\ge N_1$ and $\lVert W_n-Y\rVert_{2}<\varepsilon/2$ for $n\ge N_2$; hence $\lVert Y_n-Y\rVert_{2}<\varepsilon$ for all $n\ge\max(N_1,N_2)$. Since $\varepsilon>0$ was arbitrary, $(\lVert Y_n-Y\rVert_{2})_{n\in\mathbb{N}}$ has limit $0$ in the sense of Limit of a Sequence of Real Numbers. $\blacksquare$