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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.

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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.

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