Theorem 3

— For a numerical function $f \geq 0$ to be integrable, it is necessary and sufficient that, for every $\varepsilon > 0$, there exist an upper semi-continuous function $g \geq 0$, with finite values and compact support, and a lower semi-continuous integrable function $h$, such that $g \leq f \leq h$ and $\int (h - g)\, d|\mu| \leq \varepsilon$.

The condition is sufficient by a general criterion for integrability (§3, No. 4, Prop. 8), Prop. 5 and its Corollary 1. Let us show that the condition is necessary. If $f \geq 0$ is integrable then, for every $\varepsilon > 0$, there exists a function $u \geq 0$, continuous and with compact support, such that $N_1(f - u) \leq \varepsilon/4$. By the definition of $N_1$, this implies that there exists a lower semi-continuous function $v \geq 0$ such that $|\mu|^*(v) \leq \varepsilon/2$ and $|f - u| \leq v$. Thus, $-v(x) \leq f(x) - u(x) \leq v(x)$ for all $x \in X$, and since $u(x)$ is everywhere finite, it follows that $(u(x) - v(x))^+ \leq f(x) \leq u(x) + v(x)$ for all $x \in X$. The functions $g = (u - v)^+$ and $h = u + v$ meet the requirements.