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For a compact subset $X \subset \mathbb{R}^n$ and a continuous function $f : X \to \mathbb{C}$, $f$ is integrable on $X$ ( as both $X$ and $f$ will be bounded). Does this generalise? That is, if $X$ is a compact measure space and $f : X \to \mathbb{C}$ is a continuous function, under which circumstances (i.e. under which assumptions on the topology of $X$ and under which assumptions on the measure on $X$) is it true that $f$ is integrable on $X$? Thanks for any help!

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It works if $X$ is compact and the measure of $X$ is finite since then $$\int_X |f| d\mu \leq \int_X \max(|f|) d\mu = \max(|f|) \mu(X).$$

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  • $\begingroup$ That's fair enough! $\endgroup$
    – user
    Feb 11 at 19:11

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