I have a question regarding the conditional expectation. Any hint or help is appreciated!
Consider $a,b,$c are random variables on probability space (X, $\Sigma$, $P$),all of them are integrable and furthermore, $a$ is independent of $c$ and $b$ is also independent of $c$. Then does the following claim hold?
$$E(a|b,c) =_{a.s} E(a|b)$$
I plan to use the fact that $E(E(a|b,c)|b) = E(a|b)$, since $\sigma(b) \subset\sigma(b,c) $, but then I get stuck at showing $E(E(a|b,c)|b) = E(a|b,c)$.
Any help or hint is appreciated!