Corollary

Assume that the conditions of Prop. 9 are verified, and let $(L_\rho)_{\rho \in R}$ be a partition of $L$. Then each of the families of measures $(\mu_\lambda)_{\lambda \in L_\rho}$ admits a product measure, the family of product measures $\left( \bigotimes_{\lambda \in L_\rho} \mu_\lambda \right)_{\rho \in R}$ admits a product measure, and

This follows at once from the formulas (18) and (19) and the associativity of the product for multipliable families in $\mathbf{R}_+$ (GT, IV, §7, No. 5, Remark).

Exercises