Continuity of conditional expectation in $L_p$

Question

I'm looking at a probability space $(\Omega,\mathcal{F},P)$. Let $1\leq p<\infty$, and let $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$. I'm then asked to show that, for $X\in L_p(P)$, we have $E(X|\mathcal{G})\in L_p$ and $\| E(X|\mathcal{G})\|_p\leq \| X\|_p$. This is just a simple application of the Jensen inequality, which I have done, but I include the result for completeness.

I am then asked to prove the same for $p=\infty$, which I have also done by now.

Now, here's my problem. For $1\leq p\leq \infty$ and $(X_n)\subseteq L_p(P)$, we have $X_p\in L_p$, such that $X_n\to X$ in $L_p$. Then I am to show that $E(X_n|\mathcal{G})\to E(X|\mathcal{G})$ in $L_p$. It seems rather evident for $p=1$ and $p=\infty$. Can I simply interpolate in some manner, or is there an easy way of proving the general result?

Answer 1

Modulo results obtained above solution is very simple $$ \lim\limits_{n\to\infty}\Vert E(X_n|\mathcal{G})-E(X|\mathcal{G})\Vert_p =\lim\limits_{n\to\infty}\Vert E(X_n-X|\mathcal{G})\Vert_p \leq\lim\limits_{n\to\infty}\Vert X_n-X\Vert_p=0 $$