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I've been reading Stein's Fourier analysis book and was stuck on the following exercise:

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I was able to do parts (a) and (b) ((a) for a function with $f(x) \neq 0 \iff x = 0$ and taking balls around a nonzero point for (b)), but I wasn't able to find an argument for part (c).

For context, the chapter is about the mean-square convergence and pointwise convergence of Fourier series, but I cannot find a way to relate them with part (c).

(Also for context, the functions are real-valued!)

Thanks!

lmksdfa
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    You have to show that a Riemann's integrable function which is $0$ at the points of continuity has integral equal to $0$. Somewhere in the text there should be a characterization of Riemann's integrable functions as those functions which are bounded and continuous except on a set of measure $0$. – Salcio Jan 02 '23 at 14:03

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$f$ is Riemann integrable if and only if $D_f = \{\text{points of discontinuity of } f\}$ has measure 0. If $f^2$ is discontinuous at $x$, then $x\in D_f$ and thus $D_{f^2}\subset D_f$ so that $D_{f^2}$ has measure 0. Then $$\int_0^{2\pi}f^2 = \int_0^{2\pi}f1_{D_{f^2}}+\int_{0}^{2\pi}f1_{D_{f^2}^c}$$. The first integral on the right vanishes as $D_{f^2}$ has measure zero. The integral on the right vanishes by our hypothesis. Note we have to assume the authors mean $f$ is Riemann integrable here. Otherwise, take $f$ to be discontinuous everywhere, say $1_{\mathbb Q^c}$, as a counterexample.

Andrew
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