Proposition 7

Let X be a locally compact topological space and $\nu$ a measure on X. Let $g:\mathbf{C}\times X\rightarrow \mathbf{C}$ be a continuous function with compact support. For $z\in \mathbf{C}$, put

a) The mapping $h$ of $\mathbf{C}$ into $\mathbf{C}$ is continuous and bounded;

b) The mapping of X into $\mathscr{L}(E)$ defined by $x\mapsto g(u, x)$ is $\nu$-integrable and one has

The function $h$ is bounded because $g$ is continuous with compact support, and it is continuous by INT, IV, p. 144, § 4, n$^o3$, cor. 1. Since $g$ is continuous and with compact support, the mapping $x\mapsto g(u, x)$ is continuous from X into $\mathscr{L}(E)$ (TG, X, p. 28, th. 3 and prop. 5, c)). This mapping has compact support, hence is bounded and integrable on X with respect to $\nu$. Let us write

Let $y$ and $z$ be elements of E and $\mu$ the spectral measure of $(y, z)$ relative to $u$. One has

X X Sp($u$)

(formula (6), p. 271). Since $g\in \mathscr{K}(\mathbf{C}\times X)$, it follows that

by INT, III, p. 84, § 4, n$^o1$, th. 2 and formula (6), p. 271. This proves that $v=h(u)$, as required.