Random Variable convergence in $L^{q}$ space, then the conditional expectation also converges in $L^{q}$

Question

Let $q\geq 1$, and $X_{n}, X\in L^{q}$ are random variables in probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $X_{n}\rightarrow X$ in $L^{q}$, then for every sub $\sigma-$algebra $\mathcal{G} \subseteq \mathcal{F}$, I have $E[X_{n}|\mathcal{G}] \rightarrow E[X|\mathcal{G}]$ in $L^{q}$.

Answer 1

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $\mathcal{G}\subseteq\mathcal{F}$ be a sub $\sigma$-algebra. We recall Jensen inequality: For any convex function $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ and any random variable $\xi:\Omega\rightarrow\mathbb{R}$. If $E\left[\left|\varphi(\xi)\right|\right]<\infty$, then $\varphi\left(E\left[\xi\mid\mathcal{G}\right]\right)\leq E\left[\varphi(\xi)\mid\mathcal{G}\right]$ (a.e.).

Let $p\in[1,\infty)$ be fixed. We go to show that for any $X\in L^{p}$, we have $||E\left[X\mid\mathcal{G}\right]||_{p}\leq||X||_{p}$. Let $X\in L^{p}$ be given. Let $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ be defined by $\varphi(x)=|x|^{p}$. Clearly $\varphi$ is convex and $\varphi(X)$ is integrable. By Jensen inequality, \begin{eqnarray*} |E\left[X\mid\mathcal{G}\right]|^{p} & = & \varphi\left(E\left[X\mid\mathcal{G}\right]\right)\\ & \leq & E\left[\varphi(X)\mid\mathcal{G}\right]\\ & = & E\left[|X|^{p}\mid\mathcal{G}\right]. \end{eqnarray*} Clearly $\Omega\in\mathcal{G}$, therefore \begin{eqnarray*} \int|E\left[X\mid\mathcal{G}\right]|^{p}dP & \leq & \int_{\Omega}E\left[|X|^{p}\mid\mathcal{G}\right]dP\\ & = & \int_{\Omega}|X|^{p}dP. \end{eqnarray*} It follows that $||E\left[X\mid\mathcal{G}\right]||_{p}\leq||X||_{p}$.