Exercise 9, § 6

Let $\mu$ and $\nu$ be two diffuse positive measures on a locally compact space T. Show that, for $\nu$ to be a measure with base $\mu$, it suffices that every $\mu$-measurable subset of T be also $\nu$-measurable (argue by contradiction using Th. 3 and Prop. 13 of §5, and Exer. 8 b) of §6).