Exercise 4, § 2

Let $\mu$ be a positive measure on T, and $(\nu_i)_{i \in I}$ a summable family of positive measures on T such that $\mu \leq \sum_i \nu_i$. There exists a summable family $(\mu_i)_{i \in I}$ of measures on T such that $\mu = \sum_i \mu_i$ and $\mu_i \leq \nu_i$ for all $i \in I$. (Using Prop. 4, reduce to the case that T is compact, then to the case that $I = \mathbf{N}$. Then construct the $\mu_i$ by induction.)