Lemma 1

— *Let $X, Y, Z$ be three topological spaces, with $Y$ locally compact. Let $(x, y) \mapsto xy$ be a continuous mapping of $X \times Y$ into $Z$, which defines a mapping $x \mapsto u_x$ of $X$ into $\mathscr{F}(Y; Z)$ by the relation $u_x(y) = xy$. Let $f$ be a continuous function on $Z$ with values in $\mathbf{R}$ or in a Banach space, S the support of $f$, and $\mu$ a measure on $Y$. Assume that for every $x_0 \in X$, there exists a neighborhood $V$ of $x_0$ in $X$ such that $\bigcup_{x \in V} u_x^{-1}(S)$ is relatively compact in $Y$. Then: a) for every $x \in X$, $f \circ u_x$ is continuous on $Y$, with compact support; b) the mapping $x \mapsto \int_Y f(xy)\, d\mu(y)$, which is defined by a), is continuous on $X$.

The assertion a) is obvious. Let us prove b). Since continuity is a local property, we may reduce to the case that $\bigcup_{x \in X} u_x^{-1}(S)$ is contained in a compact subset $Y'$ of $Y$. Since the function $(x, y) \mapsto f(xy)$ is continuous on $X \times Y$, $f \circ u_x$ tends to $f \circ u_{x_0}$ uniformly on $Y'$ as $x$ tends to $x_0$

(GT, X, §3, No. 4, Th. 3), therefore $\mu(f \circ u_x)$ tends to $\mu(f \circ u_{x_0})$. Whence the lemma.

Let us now return to the previous notations.