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I'm stuck on a measure theory problem and need some hints. Let $S=[0,1]\times[0,1]$ be the unit square in $\mathbb{R}^2$ and $f\in L^1(S)$. Suppose that for any $g$ continuous on $[0,1]$ we have $\int_{0}^{1} f(x,y)g(y)dy=0$ for a.e. $x\in [0,1]$. Then $f=0$ a.e. on S. Thanks in advance.

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    I think using Weirstrass approx., or just use the case g=1 on S2011-08-24

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