I am having a little bit of a problem gaining some ground on proving the following expression:
Assume I have a function $h(x,y,z)$ and I am interested in computing its expectation conditional on $z$. I do not have much information about the properties of $h$, e.g I do not know it is continuous. Specifically, I am interested on
$$ \mathbb{E}\left[h(x,y,z) | z \right] = \int h(x,y,z) dF(y,x|z) $$
For some particular reason, it is easier for me to compute the following integral:
$$\int h(x,y,z) f_{h}(h|z,x)dF(x|z)$$
Are these two expressions the same? In other words, is it true that
$$ \int h(x,y,z) f_{h}(h|z,x)dF(x|z)=\int h(x,y,z) f_{y}(y|z,x)dF(x|z) $$
I think it must be related with the usual formulas of change of variables but I am not that sure.