Corollary 3

— Suppose that $\lambda$ (resp. $\mu$) is the sum of a summable family $(\lambda_\alpha)_{\alpha \in A}$ (resp. $(\mu_\beta)_{\beta \in B}$) of measures on $S$ (resp. $T$). The family $(\lambda_\alpha \otimes \mu_\beta)_{(\alpha, \beta) \in A \times B}$ is then summable, and its sum is $\lambda \otimes \mu$.

For, let $p$ be the encumbrance $\sum_{\alpha, \beta} (\lambda_\alpha \otimes \mu_\beta)^\bullet$; if $f \in \mathscr{F}_+(S)$ and $g \in \mathscr{F}_+(T)$, then obviously $p(f \otimes g) = \lambda^\bullet(f)\mu^\bullet(g)$. The proof of Cor. 1 then shows that $p$ is locally bounded, so that the family $(\lambda_\alpha \otimes \mu_\beta)$ is summable (§ 1, No. 7, Prop. 7). Its sum $\eta$ then satisfies $\eta^\bullet = p$ (§ 1, No. 7, Prop. 7), and Prop. 10 implies $\eta = \nu$.