Proposition 4

— For every numerical function $f \geqslant 0$ that is lower semi-continuous on $T$, $\mu^\bullet(f) = \mu^*(f)$.

For, let $g$ be a function in $\mathscr{K}_+(T)$ such that $g \leqslant f$. If $K$ is the (compact) support of $g$, then $\mu(g) \leqslant \mu^*(f \varphi_K) \leqslant \mu^\bullet(f)$. It follows, by the definition of upper integral, that $\mu^*(f) \leqslant \mu^\bullet(f)$, therefore $\mu^*(f) = \mu^\bullet(f)$ (formula (1)).