Corollary

Let $(f_n)$ be a sequence of numerical functions $\geqslant 0$ such that, for every $x \in X$, $\lim_{n \to \infty} f_n(x) = +\infty$. If $\mu$ is not the zero measure, then $\lim_{n \to \infty} \mu^*(f_n) = +\infty$.

If $f_0$ is the constant function equal to $+\infty$, then $f_0$ is the upper envelope of all the functions of $\mathscr{K}_+$, and, since $\mu \neq 0$, we have $\mu^*(f_0) > 0$; but since $f_0 = \alpha f_0$ for every $\alpha > 0$, necessarily $\mu^*(f_0) = +\infty$ (Prop. 11). The inequality (13) then shows that $\mu^*(f_n)$ tends to $+\infty$ with $n$.