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Let $p>1$. If $f=f(x,y)$ is such that $f\in L^p([a,b]\times [a,b])$, then $f_y(x)\in L^p([a,b])$ for almost all $y\in[a,b]$ and $f_x(y)\in L^p([a,b])$ for almost all $x\in[a,b]$.

Is this a consequence of Fubini's theorem?

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# Fubini's Theorem

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