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Let $K\subset[0,1]$ be compact and consider the function $1_K:$ $$ 1_K(x)=\begin{cases} 0 & \text{if } x \not\in K \\ 1 & \text{if } x \in K \end{cases} $$

My question: is $1_K$ Riemann integrable?

According to the Lebesgue's criterion for Riemann integrability, it suffices to know that if $m(A)=0$ where $m$ is the Lebesgue measure, and $A$ is the set of points of discontinuity of $1_K$.

The question is trivial when $K$ is a finite union of closed intervals. But I don't see how to deal with the general case.

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1 Answer 1

The Smith–Volterra–Cantor set is compact and its boundary (since it's closed with empty interior, the set equals its boundary) has positive Lebesgue measure, implying that the question cannot be solved in the general case.

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By "its boundary isn't null", do you mean that its measure is not $0$? –  Pratyush Sarkar Aug 9 '13 at 14:19
    
Yes, it has positive Lebesgue measure. Thanks, I've edited my answer. –  Jonathan Y. Aug 9 '13 at 14:58
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