Conditional expectations and the tower property

Question

What can I say about $E(XE(X|\mathcal{G}))$ if $X$ is $\mathcal{F}$-measurable and $\mathcal{G}$ is a sub-$\sigma$-algebra of $\mathcal{F}$. I konw that the tower property somehow kicks in but I always fail to get it. Any help?

Answer 1

You probably want to assume $X\in L^2$ (at the very least you need $\mathbb{E}[X\mid\mathcal{G}]\in L^2$). Then iterated expectation gives $$ \begin{align*} \mathbb{E}[X\mathbb{E}[X\mid\mathcal{G}]] &=\mathbb{E}[\mathbb{E}[X\mathbb{E}[X\mid\mathcal{G}]\mid\mathcal{G}]]\\ &=\mathbb{E}[\mathbb{E}[X\mid\mathcal{G}]\cdot\mathbb{E}[X\mid\mathcal{G}]]\\ &=\mathbb{E}[\mathbb{E}[X\mid\mathcal{G}]^2] \end{align*} $$ Unfortunately that is pretty much all you could say. If $X$ is real-valued, then this expectation between $(\mathbb{E}X)^2$ and $\mathbb{E}X^2$, the former achieved by having the $\mathcal{G}$ independent of $X$ (such as the trivial $\sigma$-algebra) and the latter by $\mathcal{G}=\sigma(X)$ or larger.