Continuous function on compact is integrable

Question

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!

Answer 1

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