Let $(X,\mathscr B_X)$ and $(Y,\mathscr B_Y)$ be two measure spaces and $(Z,\mathscr B_Z)$ be their product space. Consider two finite measures (not necessarily product measures) $\mu,\nu$ on $(Z,\mathscr B_Z)$. Suppose that for any $A\in \mathscr B_X$ and for any $B\in \mathscr B_X$ it holds that $$ \mu(A\times B)\geq \nu(A\times B). $$ Does it mean that $\mu(C)\geq \nu(C)$ for any $C\in \mathscr B_Z$?
Some thoughts: clearly, the question can be equivalently stated as suppose that a measure $\lambda$ on $(Z,\mathscr B_Z)$ is non-negative on rectangles. It it a non-negative measure?
I was going to apply monotone class-like arguments, but I do not know what to do with complements. Clearly, the inequality is preserved under countable disjoint unions, though.