Lemma

Let X be a locally compact space having a countable base. Then, there exists in $\mathscr{K}(X)$ a countable subset S having the following property: for every function $f \in \mathscr{K}(X)$, there exist a sequence $(f_n)$ of elements of S and a positive function $\varphi \in S$ such that, for every number $\varepsilon > 0$, $|f_n - f| \leq \varepsilon \varphi$ provided n is sufficiently large.

Let $X'$ be the Alexandroff compactification of X, which is a metrizable compact space (GT, IX, §2, No. 9, Prop. 16 and Cor.); we identify $\mathscr{K}(X)$ with a subset of $\mathscr{C}(X')$. Let $S'$ be a countable dense subset of the Banach space $\mathscr{C}(X')$ (GT, X, §3, No. 3, Th. 1); we can suppose that $S'$ contains the constant function $n$ for every $n \in \mathbf{N}$. Let $(U_n)$ be a sequence of relatively compact open sets in X, with union X, such that $\overline{U}_n \subset U_{n+1}$ for all n (GT, I, §9, No. 9, Prop. 15), and let $\varphi_n$ be a function in $\mathscr{K}_+(X)$ equal to 1 on $\overline{U}_n$. We denote by S the countable set of elements of $\mathscr{K}(X)$ of the form $\varphi_n g$ ($n \in \mathbf{N}, g \in S'$). If $f \in \mathscr{K}(X)$, let $(g_n)$ be a sequence of elements of $S'$ that converges uniformly to $f$, and let k be an integer such that the support of $f$ is contained in $U_k$. Finally, let m be an integer that is an upper bound for the norms of the functions $g_n$. The functions $f_n = \varphi_k g_n$ belong to S and satisfy the statement, with $\varphi = m \varphi_k$.

The lemma having been established, and the mapping $t \mapsto \lambda_t(g)$ being essentially $\mu$-integrable for every $g \in S$, the mapping $t \mapsto (\lambda_t(g))_{g \in S}$ of T into $\mathbf{R}^S$ is $\mu$-measurable (Ch. IV, §5, No. 3, Th. 1). The set $\mathscr{K}$, of compact subsets K of T such that the restriction of this mapping to K is continuous, is therefore $\mu$-dense, and it will suffice to show that the restriction of $\Lambda$ to every $K \in \mathscr{K}$ is continuous. Now, let $f$ be any element of $\mathscr{K}(X)$, $f_n$ and $\varphi$ elements of S satisfying the statement of the Lemma; the function $t \mapsto \lambda_t(f)$ is then the uniform limit on K of the continuous functions $t \mapsto \lambda_t(f_n)$; it is therefore continuous on K, and the proposition is proved.