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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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