Let $X,Y : \Omega \rightarrow \mathbb{R}$ be two random variables on a probability space $(\Omega , \Sigma, P)$
The expectation value of $f(X,Y)$ is then given by
$\int_{\Omega} f(X( \omega) , Y(\omega) ) dP(\omega).$
Now is this the same as
$\int_{\mathbb{R}^2} f(x,y) dP_{(X,Y)}(x,y)$ or the same as $\int_{(X,Y)(\Omega)} f(x,y) dP_{(X,Y)}(x,y)$ here $dP_{(X,Y)}$ denotes the push-forward measure. $(X,Y)(\Omega):=\{(X(\omega),Y(\omega)); \omega \in \Omega\}.$