Let $X$ and $Y$ two independent random variables with distribution $F_X$ and $F_Y$. Consider a function $g(X,Y) \in R$. Let $E_P\{\cdot\}$ denote expectation under the distribution $P$.
Am I correct that \begin{equation} E\{g(X,Y)\} = E_{F_Y}\{ E_{F_X} g(X,Y)\} \end{equation}
On the other end assuming that $X$ and $Y$ are not independent, and that $F_{X|Y}$ is the conditional distribution of $X|Y$, am I correct that \begin{equation} E\{g(X,Y)\} = E_{F_Y}\{ E_{F_{X|Y}} \{ g(X,Y) | Y\} \} \end{equation}