Lemma 3

— Let X be a topological space (Hausdorff or not), f a lower semi-continuous function $\geqslant 0$ defined on X; then f is the limit of an increasing sequence $(f_n)_{n \in \mathbf{N}}$ of lower semi-continuous functions on X, such that each function $f_n$ is a linear combination, with positive coefficients, of characteristic functions of open sets.

Given two integers $k \geqslant 1$ and $n \geqslant 1$, let us denote by $J_{kn}$ the characteristic function of the interval $]k/2^n, +\infty]$ of $\overline{\mathbf{R}}$. For every $x \in \overline{\mathbf{R}}_+$, set

it is immediate that the sequence $(u_n(x))_{n \geqslant 1}$ is increasing and admits x as limit. The sequence of functions $f_n = u_n \circ f$ is therefore increasing and converges to $f$, and one has $f_n = 2^{-n} \sum_{k=1}^{n \cdot 2^n} \varphi_{U(k,n)}$, where $U(k,n)$ is the open set $f^{-1}([k/2^n, +\infty])$ of $X$.

Let us pass to the proof of b). The function $I_f$ being the upper envelope of the increasing directed family of functions $I_f^L$, where $L$ runs over the set of compact subsets of $T$ (Lemma 2), it will suffice to show that the functions $I_f^L$ are lower semi-continuous; the formula (9) may then be deduced from (6) by passing to the upper envelope over $L$ (§ 1, No. 6, Prop. 5).

Thus let $\mathscr{H}$ be the set of positive lower semi-continuous functions $f$ on $S \times T$ such that $I_f^L$ is lower semi-continuous for every compact subset $L$ of $T$. By Prop. 5 of § 1, No. 6, the supremum of every increasing directed set of elements of $\mathscr{H}$ belongs to $\mathscr{H}$. By Lemma 3, it will therefore suffice to prove that the characteristic function of an open set $W$ of $S \times T$ belongs to $\mathscr{H}$. Moreover, by the definition of the product topology on $S \times T$, the open set $W$ is the union of an increasing directed family $(W_\alpha)_{\alpha \in A}$ of open sets of the form

where the $U_i$ are open in $S$ and the $V_i$ are open in $T$; by the remarks made above, it will suffice to show that the characteristic function of such an open set belongs to $\mathscr{H}$. Let then $s \in S$, and let $U$ be the intersection of the family (possibly empty) formed by the open sets $U_i$ containing $s$; one sees immediately that $\varphi_W(s,t) \leq \varphi_W(s',t)$ for all $s' \in U$ and $t \in T$, whence, by integration, $I_{\varphi_W}^L(s) \leq I_{\varphi_W}^L(s')$ for all $s' \in U$. Consequently $I_{\varphi_W}^L$ is lower semi-continuous, and the proposition is established.