Take Fubini's theorem as stated above. Say there exists $(M,\mathscr{M}, \mu)$ and $(N,\mathscr{N},\nu)$, then the product measure is such that $\pi(A\cap B)=\mu(A)\nu(B)$ where $\pi$ is defined over the measurable space $(M \times N,\sigma(\{J\times N|J,N \in M,N \text{ respectively}\}))$. Note that the product measure is just one form of joint measure and is the case wherein there is independence between the two measures (e.g., this is clearly the case when speaking of measures induced by random variables).
Now, if we want to extend Fubinis to general cases of joint density, how does it follow? Indeed, from elementary probability, one can always change the bounds of integration and then the order of integration (by inducing Fubini's theorem) and thus I reckon one could make an argument from conditional distributions (and then introduce a notion of conditional independence where a product measure could suffice).