Suppose $X$ and $Y$ are two real-valued random variables, and $f:\mathbb{R}^2\to \mathbb{R}$ is Borel measurable.
I was wondering if $X$ and $Y$ being uncorrelated or independent implies that $$ \mathrm{E}_{X,Y} f(X,Y) = \mathrm{E}_X [\mathrm{E}_Y f(X,Y)] = \mathrm{E}_Y [\mathrm{E}_X f(X,Y)]? $$ Thanks and regards!