Proposition 7

— If $f$ is a function $\geqslant 0$ defined on $X$ such that $|\mu|^*(f) < +\infty$, then $f(x)$ is finite almost everywhere.

For, let $N$ be the set of points $x \in X$ such that $f(x) = +\infty$; for every integer $n$, $n \varphi_N \leqslant f$, whence $n |\mu|^*(\varphi_N) \leqslant |\mu|^*(f)$; since $n$ is arbitrarily large, $|\mu|^*(N) = 0$.

However, even if $X$ is compact, a function $f \geqslant 0$ defined on $X$ and everywhere finite can have infinite upper integral, as is shown by the example $X = [0, 1]$, $f(x) = 1/x$ for $x > 0$ and $f(0) = 0$, $\mu$ being Lebesgue measure on $X$.