Need help in question for conditional expectation

Question

Let $(X, \Sigma, P)$ be the probability space, let $X$ and $Y$ be a integrable random variables (all are bounded) on this space and $\Sigma_{0}$ be a sub-$\sigma$-algebra of $\Sigma$. Show that $$ E(E(X|\Sigma_{0})^2) =E(XE(X|\Sigma_{0})) $$ and $$ E(YE(X|\Sigma_{0})) =E(XE(Y|\Sigma_{0})) $$

I have try to use LIE but don't know how to proceed this question, any help or hint is extremely appreciated!

Answer 1

Both are consequences of the following fact:

For every square integrable $X$ and $Y$, $E(YE(X\mid\Sigma_0))=E(E(Y\mid\Sigma_0)E(X\mid\Sigma_0))$ almost surely.

Hence your task is twofold:

• Prove the fact above.
• Show that it implies both results in your question.

To prove the fact above, one can first show that $E(Y\mathbf 1_B)=E(E(Y\mid\Sigma_0)\mathbf 1_B)$ for every $B$ in $\Sigma_0$, then try to expand the class $\mathfrak C$ of random variables $Z$ measurable with respect to $\Sigma_0$ such that $E(YZ)=E(E(Y\mid\Sigma_0)Z)$. Note that if $\mathfrak C$ contains every square integrable random variable $Z$ measurable with respect to $\Sigma_0$, then the proof is over.

Answer 2

I think that you can answer both of these questions by using the fact that the operator

$ X \mapsto E[X|\Sigma_0] $

on the Hilbert space $L^2(\Omega,P)$ is the orthogonal projection onto the subspace of random variables that are measurable with respect to $\Sigma_0$. For example, the second result boils down to the fact that orthogonal projection operators are self-adjoint.

However, this is a functional analysis argument and not a probability one, so maybe you should look in Billingsley. I'd be happy to elaborate if any of the above doesn't make sense.