Proposition 7

Let X, Y, Z be locally compact spaces, $\varphi$ a continuous mapping of $X \times Y$ into Z, and $\lambda, \mu$ measures on X, Y. Assume that for every $x \in X$, $\varepsilon_x$ and $\mu$ are $\varphi$-convolvable. For $\lambda$ and $\mu$ to be $\varphi$-convolvable, it is necessary and sufficient that the mapping $x \mapsto \varepsilon_x * |\mu|$ of X into $\mathscr{M}(Z)$ be scalarly $\lambda$-integrable for the topology $\sigma(\mathscr{M}(Z), \mathscr{K}(Z))$, in which case $\lambda * \mu = \int_X (\varepsilon_x * \mu) d\lambda(x)$.

Suppose that $\lambda$ and $\mu$ are $\varphi$-convolvable. For every $f \in \mathscr{K}(Z)$, $f \circ \varphi$ is $(|\lambda| \otimes |\mu|)$-integrable, therefore the function $x \mapsto \int_Y f(\varphi(x, y)) d|\mu|(y) = \langle f, \varepsilon_x * |\mu| \rangle$ (which by hypothesis is defined for all $x \in X$) is $\lambda$-integrable; thus $x \mapsto \varepsilon_x * |\mu|$ is scalarly $\lambda$-integrable for $\sigma(\mathscr{M}(Z), \mathscr{K}(Z))$, and

whence $\lambda * \mu = \int_X (\varepsilon_x * \mu) d\lambda(x)$. Conversely, suppose that the mapping $x \mapsto \varepsilon_x * |\mu|$ of X into $\mathscr{M}(Z)$ is scalarly $\lambda$-integrable for $\sigma(\mathscr{M}(Z), \mathscr{K}(Z))$. Let $f \in \mathscr{K}_+(Z)$. Then the function $(x, y) \mapsto f(\varphi(x, y))$ is continuous and (Ch. V, §8, No. 3, Prop. 5)

Therefore $f \circ \varphi$ is $(\lambda \otimes \mu)$-integrable, so that $\lambda$ and $\mu$ are $\varphi$-convolvable.