Corollary 1

— For a finite numerical function $f \geqslant 0$, upper semi-continuous on X, to be integrable, it is necessary and sufficient that $\int^* f d|\mu| < +\infty$.

For, if $|\mu|^*(f) < +\infty$, then there exists a lower semi-continuous function $h$ such that $f \leqslant h$ and $|\mu|^*(h) < +\infty$; $h - f$ is everywhere defined and lower semi-continuous, and $|\mu|^*(h - f) \leqslant |\mu|^*(h) < +\infty$; therefore $h - f$ is integrable, and since $f(x) = h(x) - (h(x) - f(x))$ almost everywhere, $f$ is integrable.