Properties of conditional expectation, independence

Question

Any advice would be appreciated!

Is it true that for $(Ω, \mathcal F, P)$, $\mathcal G \subset \mathcal F$, if $X$ is independent of $\mathcal G$ then $E[X|\mathcal G] = E[X]$, would$E[X^2|\mathcal G] = E[X^2]$ hold?

I am unsure if this is a property of conditional expectations I missed out.

Thank you!

Answer 1

If $X$ is independent of $\mathcal{G}$ then $f(X)$ is also independent of $\mathcal{G}$ for every Borel function $f$. This is easy to verify from the fact that $\sigma(X)$ is independent of $\mathcal{G}$, and that $(f(X))^{-1}(B) = X^{-1}(f^{-1}(B))$.

In particular we have $E[f(X) \mid \mathcal{G}] = E[f(X)]$ whenever $E[f(X)]$ exists. Apply this with $f(x) = x^2$.

Answer 2

If $X$ is independent of $\mathcal G$ then $X^{2}$ is also independent of $\mathcal G$ so the answer is YES.