Riemann integrability of indicator function of compact subset of a closed interval

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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By "its boundary isn't null", do you mean that its measure is not $0$? – Pratyush Sarkar Aug 9 '13 at 14:19