Proposition 14

Let $f \in \mathscr{F}_+(T)$. If $f$ is $\mu$-moderated, then $\mu^*(f) = \mu^\bullet(f)$; if $f$ is not $\mu$-moderated, then $\mu^*(f) = +\infty$.

If $\mu^*(f) < +\infty$, there exists a lower semi-continuous function $g \geq f$ such that $\mu^\bullet(g) < +\infty$. For every $n \in \mathbf{N}$, let $G_n$ be the set of $t \in T$ such that $g(t) > 1/n$; the set $G_n$ is open, one has $\mu^\bullet(G_n) \leq n \mu^\bullet(g) < +\infty$, and $f$ is zero outside the union of the $G_n$: the function $f$ is therefore moderated.

Next, let us show that $\mu^*$ and $\mu^\bullet$ have the same value for moderated functions. Since $\mu^*$ and $\mu^\bullet$ are encumbrances, it suffices to establish the relation $\mu^*(f) = \mu^\bullet(f)$ when $f$ is a positive function, bounded above by a constant M, and zero outside an open set G of finite measure, which we shall now do.

The measure $\mu$ is the supremum, in $\mathscr{M}(T)$, of an increasing directed family $(\mu_i)_{i \in I}$ of measures with compact support: this follows at once from Prop. 9 of No. 8. Let $g$ be a lower semi-continuous function on T, between $f$ and the lower semi-continuous function $M \varphi_G$. Set $\nu_i = \mu - \mu_i$; then $\mu^\bullet = \mu_i^\bullet + \nu_i^\bullet$ (No. 2, Remark 1), consequently

One has $\nu_i^\bullet(M \varphi_G) = \mu^\bullet(M \varphi_G) - \mu_i^\bullet(M \varphi_G)$ and $\mu^\bullet(M \varphi_G) = \sup \mu_i^\bullet(M \varphi_G)$ (No. 7, Prop. 6); the number $\nu_i^\bullet(M \varphi_G)$ may therefore be made arbitrarily small by a suitable choice of $i$. Thus everything comes down to showing that one can find, for any number $c > 0$ and any index $i \in I$, a lower semi-continuous function $g$ between $f$ and $M \varphi_G$, such that $\mu_i^\bullet(g) - \mu_i^\bullet(f) \leq c$. Now, let L be the compact support of the measure $\mu_i$, and let $\lambda$ be the measure $(\mu_i)_L$; since $\mu_i$ is concentrated on L, one has $\mu_i^\bullet(h) = \mu_i^\bullet(h \varphi_L) = \lambda^\bullet(h_L)$ for every function $h \in \mathscr{F}_+(T)$ (No. 1, Lemma 1 and No. 2, Prop. 2); therefore

¹ The cited appendix in TG does not appear in GT (which translated an earlier edition of Ch. IX). Lindelöf spaces are defined in GT, I, §9, Exer. 14; this exercise and the definition of Souslin space (GT, IX, §6, No. 2, Def. 2) cover the material needed here.

But L is compact; therefore $\lambda^* = \lambda^*$, consequently there exists a lower semi-continuous function $h$ defined on L, that is $\geq f_L$ and is such that $\lambda^*(h) \leq \lambda^*(f_L) + c$. Since the set L is closed in T, the function k equal to h on L, and to $+\infty$ on $T - L$, is lower semi-continuous on T and is $\geq f$, and $\lambda^*(k_L) = \lambda^*(h) \leq \lambda^*(f_L) + c$. It remains only to set $g = \inf(k, M \varphi_G)$: $g$ is lower semi-continuous, $g \geq f$ and