I feel that the following corollary of the Monotone Class Theorem should appear somewhere in the literature, but I haven't been able to find it any of the measure theory books that I have checked.
Lemma: If $\nu$ is a signed measure on the product space $\Omega_1 \times ... \times \Omega_n$, and $\nu(A_1 \times ... \times A_n)\ge0$ for all measurable $A_1\subset\Omega_1,...,A_n\subset\Omega_n$, then $\nu(A)\ge0$ for all measurable $A\subset\Omega_1 \times ... \times \Omega_n$.
Does anyone know where to find it?