# A reference for “if a signed measure on a product space is nonnegative on rectangles, then it is nonnegative”

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?

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If I were writing a textbook, I'd make this an exercise. I wouldn't expect to find it in a textbook per se, and I don't think it would be necessary to give a reference if using this fact in a paper. I'd just say something like: "The measure is nonnegative on rectangles, so by a monotone class argument, it is nonnegative." Anyone who's ever used the monotone class theorem should see immediately how it follows. – Nate Eldredge Jan 23 '12 at 19:04