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Let $A\subseteq \mathbb{R}^n$ bounded set. I have to show that $\chi _A$ is Riemann integrable iff $m(\partial A)=0$. It's a question from a text book i couldn't think of the answer to...

Michael
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    Are you permitted to use the characterization of Riemann-integrable functions as those that are bounded and whose points of discontinuity form a set of measure zero? If not, perhaps consult a proof of that theorem and simplify it to adapt it to the case of a characteristic function. – user119191 Jan 05 '14 at 14:30

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For any partition of a hypercube conaining $A$, the difference between upper and lower Riemann sum for $\chi_A$ is at least $m(\partial A)$. Hence $m(\partial A)=0$ is necessary for Riemann integrability.

If $m(\partial A)$ is zero, a sufficiently fine partition of a hypercube containing $A$ has only a negligible portion of subhypercubes affected by the boundary, so that upper and lower Riemann sum differ by an arbitrarily small amount.

Hagen von Eitzen
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