Equivalent condition on marginals

Question

Let $\mu$ be a probability measure on $X$ (compact), $v$ a probability measure on $Y$ (compact), and let $\gamma$ be a probability measure on $X \times Y$: if $$\forall \psi \in C(X), \int_X \psi(x)d\mu=\int_{X \times Y} \psi(x)d\gamma$$ then $(\pi^x)_{*}\gamma=\mu$ (so, $\mu(\cdot)=\gamma((\pi^{x})^{-1}(\cdot))$, where $\pi^x$ represents the projection on $X$. The statement is analogous, mutatis mutandis, with respect to $Y$ and $v$.
I have been able to prove the opposite direction of the statement by a usual chain of reasoning in measure theory but I am stuck on this direction.
At a certain point I'd like to obtain something like "for all B measurable in X, then $\mu(B)=\gamma(B\times Y)$", but I cannot insert immediately an indicator function in the integrals since it would not satisfy the continuity on $X$ assumption.