Equality of expectations if identically distributed

Question

Let $X,Y,Z$ be independent random variables on some probability space $(\Omega, P)$ with values in $\mathbb R$ such that $X \sim Y$, i.e. $X$ and $Y$ have the same distribution. Let $f \colon \mathbb R^2 \to \mathbb R$ be a measurable function. Is it always true that $$\mathbb E [ f(X,Z) ] = \mathbb E[f(Y,Z) ].$$ If not, are there any sufficient conditions on $f$ for this to be true? E.g. when $f(x,y) = xy$ this is true since $$\mathbb E[XZ] = \mathbb E[X] \mathbb E[Z] = \mathbb E[Y] \mathbb E[Z] = \mathbb E[YZ]$$ by independence. However, I was not able to prove the more general result. Any help is appreciated!

Answer 1

If $f(X,Z)$ and $f(Y,Z)$ are integrable then they gave the same expectation . The common value is $\iint f(x,z) dF_X(x)dF_Z(z)$. The equation also holds if $f$ is a non-negative measurable function. This follows from the fact that $(X,Z)$ and $(Y,Z)$ have the same two dimensional distribution.