Conditional Expectation.

Question

1. Let $X$ be a random variable in $L^2(\Omega, \Sigma, P)$ and $\mathcal G$ a sub-$\sigma$-algebra of $\Sigma$.
   Prove that $E[(X-E[X\mid\mathcal G])^2] \le E[(X-E[X])^2]$.

As conditional expectation can be viewed as a projection of $X$ from $L^2(\Sigma)$ into a subspace $L^2(\mathcal G)$, the norm $E[(\bullet)^2]$ would be smaller. And the inequality is then obvious. However, this is rather intuitive. I have some trouble in giving a rigorous proof. Can anyone help?

Also I am wondering if $E[|X-E[X\mid\mathcal G]|] \le E[|X-E[X]|]$. This seems true from the above reasoning?

2. Let $T$ and $S$ be stopping times and $\mathcal F_T$ and $\mathcal F_S$ be the stopping time $\sigma$-algebra.
   Prove that $E[E[Y\mid\mathcal F_S]\mid\mathcal F_T] = E[E[Y\mid\mathcal F_T]\mid\mathcal F_S] = E[Y\mid\mathcal F_{T\land S}]$.

It seems obvious that the smaller stopping time is the one that matters. But still, I am not able to give the proof.

Thanks a lot!

Answer 1

The key is to utilize to definition of conditional expectation and its properties, which is based on your thorough understanding of the difference between conditional expectation in advanced probability (based on measure theory) and that in elementary probability (based on calculus).

The solution to the first question.

The solution to the second question.