Given $X, ~Y$ random variables on probability space $(\Omega, \mathcal{F},\mathbb{P})$ with $\mathbb{E}(X^2), \mathbb{E}(Y^2) < \infty $. Let $Z=\mathbb{E}[Y \mid \mathcal{G}]$ where $\mathcal{G} \subset \mathcal{F}$ is a $\sigma$-algebra.
I want to show that $\mathbb{E}(|XZ|)<\infty$.
I know that I have to use the fact that $\mathbb{E}(X^2),\mathbb{E}(Y^2) < \infty$ but I don't know how. $\mathbb{E}(|XZ|) = \mathbb{E}(|X \mathbb{E}(Y \mid \mathcal{G})|) \cdots $
Do you have any hints for me? I don't think that the Jensen's inequality helps, and the problem is that I don't know whether $X, Y$ are independent or not.
By the Cauchy-Schwarz inequality, we have $$|E(XZ)|\le \sqrt{E|X|^2}\sqrt{E|Z|^2},$$ and the Jensen's inequality for conditional expectations gives us $$E|Z|^2=E|E(Y|G)|^2\le E[E(Y^2|{\cal G})]=E[Y^2]<\infty. $$