Lemma about the integral of a function with compact support

Lemma 16.4 (p. 140) of Munkres’ Analysis on Manifolds says:

Let $A$ be open in $\mathbb{R}^n$; let $f: A \rightarrow \mathbb{R}$ be continuous. If $f$ vanishes outside a compact subset $C$ of $A$, then the integrals $\int_A f$ and $\int_C f$ exist and are equal.

The first step in his proof is saying that the integral $\int_C f$ exists because $C$ is bounded and $f$ is continuous and bounded on all of $\mathbb{R}^n$.

But don’t you need $C$ to be rectifiable (i.e. bounded and boundary has measure $0$) for integrability? The fat cantor set is compact but not rectifiable, so the integral over it won’t exist.

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That depends on the theory of integration that you have at your disposal. Integration based on Jordan measure (i.e., multiple Riemann integral) indeed runs into trouble when sets have “fat” boundaries. But for Lebesgue integral the boundary is not an issue. Since every Borel set is Lebesgue measurable, we can integrate any bounded Borel function over any bounded Borel set. This covers the case of $f$ being integrated over $C$.

I think that Theorem 13.5 of the same book may be of use here. The conditions set out in the lemma imply that $f$ is continuous at $\partial C$. Since it vanishes outside $C$, the limit must be $0$ at all points on $\partial C$. Therefore, $f$ and $C$ satisfy the conditions of Theorem 13.5.

It is a bit strange that Munkres didn’t offer more explanation, seeing that he is far from terse for most of the prior material. On the other hand, the argument above could probably be considered standard at this stage of the book.