I am not a mathematician, so I am sorry if this question is too easy or some notational detail is not correct. I am trying my best!
I have got a random Variable $X$ in $\mathbb{R}^N$ with pdf $p(X)$ and a continuous, smooth function $g:\mathbb{R}^N \rightarrow \mathbb{R}$. I am interested in finding the pdf of $Y=g(X)$ (that is the probability of a point $x$ with $g(x)=y$) but as an integral over $X$.
I think the solution should look like
$$ p(Y=y) = \int_{\mathbb{R}^n} \delta(g(x)-y) p(X=x) \, dx \enspace, $$
where $\delta(x)$ is the dirac-delta function.
But I can not proof it and i do not know of any easy to understand reference. Does someone here have any hint?