# almost everywhere zero in product measure space

Let $$M_{1}$$ and $$M_2$$ be measure spaces with measures $$\mu_{1}$$ and $$\mu_{2}$$ respectively, and $$f:M_{1}\times M_{2}\rightarrow \mathbb{R}$$ be a measurable function (I assume $$M_{1}\times M_{2}$$ is equipped with product measure).

Suppose there exists a measurable set $$N\subset M_{2}$$ with $$\mu_{2}(N)=0$$ such that for any fixed $$y\in M_{2}-N$$, the function $$g(x)=f(x,y)$$ is almost everywhere zero with respect to measure $$\mu_{1}$$. Is it necessarily true that $$f(x,y)=0$$ almost everywhere with respect to the product measure?

I think it is true but I kind of have difficulty proving this statement as it may involve uncountable union. Thanks.

Let $$E=\{(x,y): f(x,y) \neq 0\}$$. Fro any $$y\in M_2$$ the section $$E^{y}$$ is $$\{x:f(x,y)=0\}$$ and its $$\mu_1$$ measure is $$0$$ by hypothesis. By Fubini's Theorem this implies $$(\mu_1\times \mu_2) (E)=0$$ which is what we have to prove.