Corollary

— For every integrable (resp. integrable and $\geq 0$) numerical function $f$, there exist an increasing sequence $(g_n)$ of upper semi-continuous functions that are integrable (resp. integrable, with finite values $\geq 0$, and with compact support), and a decreasing sequence $(h_n)$ of lower semi-continuous integrable functions, such that: $1^\circ$ $g_n(x) \leq f(x) \leq h_n(x)$ for all $x \in X$ and every integer $n$; $2^\circ$ $f(x)$ is equal almost everywhere to the lower envelope $h$ of the sequence $(h_n)$ and to the upper envelope $g$ of the sequence $(g_n)$; $3^\circ$ $\int f\, d\mu = \lim_{n \to \infty} \int g_n\, d\mu = \lim_{n \to \infty} \int h_n\, d\mu$.

Suppose first that $f \geq 0$. By Th. 3 there exist, for every $n$, a lower semi-continuous integrable function $v_n$, and an upper semi-continuous function $u_n \geq 0$ with finite values and compact support, such that

setting $g_n = \sup(u_1, u_2, \ldots, u_n)$ and $h_n = \inf(v_1, v_2, \ldots, v_n)$, the sequences $(g_n)$ and $(h_n)$ meet the requirements. For, since $g \leq f$, $g$ is integrable by Prop. 4 of No. 3, and since

we have (No. 3, Prop. 4), which proves that $f$ and $g$ are equivalent. One argues similarly for the sequence $(h_n)$.

If $f$ is not positive, we can apply the foregoing to $f^+$ and $f^-$, thus there are two increasing sequences $(g'_n), (g''_n)$ of upper semi-continuous integrable functions, and two decreasing sequences $(h'_n), (h''_n)$ of lower semi-continuous integrable functions, such that: $1^\circ \ g'_n \leq f^+ \leq h'_n,\ g''_n \leq -f^- \leq h''_n;$ $2^\circ \ f^+$ (resp. $-f^-$) is equal almost everywhere to the upper envelope of the $g'_n$ and to the lower envelope of the $h'_n$ (resp. to the upper envelope of the $g''_n$ and to the lower envelope of the $h''_n$); and $3^\circ$:

Moreover, we can suppose that the $g'_n$ and the $h''_n$ are everywhere finite; it is then clear that the sequences $g_n = g'_n + g''_n$ and $h_n = h'_n + h''_n$ meet the requirements.