Lemma 5

Let X be a locally compact topological space and $\nu$ a positive measure on X. On every bounded subset of $\mathscr{C}_b(X)$, the topology induced by the weak topology $\sigma (L^{\infty}(X),L^1(X))$ is coarser than the topology of compact convergence.

Let B be a bounded subset of $\mathscr{C}_b(X)$ and let $M\in \mathbf{R}_+$ be such that $\|\varphi \|_{\infty}\leqslant M$ for every $\varphi \in B$. Let $\psi \in L^1(X, \nu )$ and $\varepsilon >0$. Fix a compact subset K of X such that

For all $\varphi_1$ and $\varphi_2$ in B, we then have

whence

$|\langle \varphi_1-\varphi_2, \psi \rangle |\leqslant$ sup$_{x\in K}|\varphi_1(x)-\varphi_2(x)| \|\psi \|_1+ 2M\varepsilon$,

and the lemma follows.