Disintegration and Measure-zero sets

Question

Suppose that $X,Y$ are Radon spaces, $\mu$ is a Borel probability measure on $X$, and $\nu$ is a $\sigma$-finite Borel measure on $Y$. Fix some non-empty $A\subseteq X\times Y$ such that
$$
\mu\otimes \nu (A)=0.
$$
Can we then conclude (somehow by disintegration) that the projection of $A$ onto $Y$ are of $\nu$-measure $0$?

Answer 1

Well, if $A=N\times B,$ where $N$ is a $\mu$-null set and $B$ is any Borel subset of $Y$, then $\mu\otimes \nu(A)=0,$ but the measure of $B$ can be whatever we want.