Expectation of conditional expectation of function

Question

What are the necessary and sufficient conditions on $f$ such that \begin{align*} \mathbb{E}[\mathbb{E}[f(X)|Y]]=\mathbb{E}[f(X)] \end{align*} holds? Is it always true?

Answer 1

Suppose we have an integrable random variable $X$ on the probability space $(\Omega,\mathscr{F},P)$. If we are given a sub sigma-algebra $\mathscr{G}$ of $\mathscr{F}$, we define the conditional expectation $Y=E[X|\mathscr{G}]$ to be the almost surely unique integrable random variable such that $\int_GY=\int_GX$ for all $G$ measurable with respect to $\mathscr{G}$. Now, using that $\Omega\in \mathscr{G}$ by the definition of a sigma-algebra, we immediately get that $\int_{\Omega}E[X|\mathscr{G}]=\int_{\Omega}X$, that is, $E[E[X|\mathscr{G}]]=E[X]$. So as long as $f$ is such that the random variable $f(X)$ is integrable so that the conditional expectation can be defined, the statement is true.