Equality of Borel measures

Question

Suppose that $f\in C_c(\mathbb{R}^N)$ is an arbitrary compactly supported continuous function, such that the integral equality $$\int_{\mathbb{R}^N\times\mathbb{R}^M}f(x)d\mu(x,y)=\int_{\mathbb{R}^N}f(x)d\nu(x)$$ holds. Here $\mu$ and $\nu$ are Borel measures on their respective spaces. Is it necessarily true that $\nu(B)=\mu(B\times\mathbb{R}^M)$ for all Borel sets $B\in\mathcal{B}(\mathbb{R}^N)$? It looks like it is true, by the mollification argument of characteristic function, but I am not fully convinced.