Proposition 5

For every measure $\lambda_0 \in \mathscr{M}(X; \mathbf{C})$, the mapping $\mu \mapsto \lambda_0 \otimes \mu$ of $\mathscr{M}(Y; \mathbf{C})$ into $\mathscr{M}(X \times Y; \mathbf{C})$ is vaguely continuous.

For every function $f \in \mathscr{K}(X \times Y; \mathbf{C})$, we know that the function $h(y) = \int f(x, y) d\lambda_0(x)$ belongs to $\mathscr{K}(Y; \mathbf{C})$ (No. 1, Lemma 2), and $\langle f , \lambda_0 \otimes \mu \rangle = \langle h , \mu \rangle$, whence the proposition.