Proposition 5

— For a numerical function $f \geqslant 0$ (finite or not), lower semi-continuous on X, to be integrable, it is necessary and sufficient that $\int^* f d|\mu| < +\infty$.

It all comes down to proving that the condition is sufficient. The definition of $|\mu|^*(f)$ (§ 1, No. 1, Def. 1) proves that, for every $\varepsilon > 0$, there exists a continuous function $g \geqslant 0$, with compact support, such that $g \leqslant f$ and $|\mu|^*(f) \leqslant |\mu|(g) + \varepsilon$. But $f - g$ is lower semi-continuous and $\geqslant 0$, therefore (§ 1, No. 1, Th. 2)

in other words $N_1(f - g) = |\mu|^*(f - g) = |\mu|^*(f) - |\mu|(g) \leqslant \varepsilon$, which proves that $f$ is integrable (§ 3, No. 4, Prop. 7).