Lemma 3

— Let T and X be two locally compact spaces with countable bases, $\mu$ a positive measure on T, and $t \mapsto \lambda_t$ ($t \in T$) a family of positive measures on X. If the mapping $t \mapsto \lambda_t$ is scalarly $\mu$-integrable (for the topology $\sigma(\mathscr{M}(X), \mathscr{K}(X))$), then the family $t \mapsto \lambda_t$ is $\mu$-adequate ($§ 1$, No. 1, Example).

For, Lemma 1, applied to $\mathscr{K}(X)$, shows that the mapping $t \mapsto \lambda_t$ is vaguely $\mu$-measurable ($§ 1$, No. 5, Prop. 13).