Theorem 1

Let $H$ be a nonempty set of functions in $\mathscr{J}_+$, directed for the relation $\leq$. For every positive measure $\mu$ on $X$,

Let $f = \sup_{g \in H} g$. We shall first prove the theorem for the special case that the functions $g \in H$ and their upper envelope $f$ belong to $\mathscr{H}_+$. It then follows from Dini’s theorem (GT, X, §4, No. 1, Th. 1) that the section filter of $H$ converges uniformly to $f$ on every compact subset of $X$, and in particular on the support K of $f$. Since $0 \leq g \leq f$ for every function $g \in \mathbf{H}$, the support of every function in $\mathbf{H}$ is contained in $K$; but by definition $\mu$ is continuous on the vector space $\mathscr{K}(X, K; \mathbf{C})$ of continuous functions with support contained in $K$, for the topology of uniform convergence; whence the relation (2) in this case.

Let us pass to the general case. It is clear that $\mu^*(g) \leq \mu^*(f)$ for every function $g \in \mathbf{H}$. By Def. 1, it all comes down to showing that, for every function $\psi \in \mathscr{K}_+$ such that $\psi \leq f$,

For every function $g \in \mathbf{H}$, let $\Phi_g$ be the set of functions $\varphi \in \mathscr{K}_+$ such that $\varphi \leq g$, and let $\Phi$ be the union of the sets $\Phi_g$ as $g$ runs over $\mathbf{H}$; since $\mathbf{H}$ is directed, so is $\Phi$, and $f = \sup_{\varphi \in \Phi} \varphi$. Since $\psi \leq f$, $\psi$ is the upper envelope of the set of functions $\inf(\psi, \varphi)$ as $\varphi$ runs over $\Phi$; but since $\psi$ and the functions $\inf(\psi, \varphi)$ belong to $\mathscr{K}_+$, the first part of the proof shows that $\mu(\psi) = \sup_{\varphi \in \Phi} \mu(\inf(\psi, \varphi))$. Now, each $\varphi \in \Phi$ belongs to a set $\Phi_g$, therefore

from which it follows at once that $\mu(\psi) \leq \sup_{g \in \mathbf{H}} \mu^*(g)$. We have thus proved that $\mu^*(f) = \sup_{g \in \mathbf{H}} \mu^*(g)$; the relation $\mu^*(f) = \lim_{g \in \mathbf{H}} \mu^*(g)$ is then a consequence of the monotone limit theorem (GT, IV, §5, No. 2, Th. 2).