Theorem 2 (Lebesgue–Nikodym)

— Let $\mu$ and $\nu$ be two positive measures on $T$. The following properties are equivalent:

1) $\nu$ is a measure with base $\mu$. 2) Every locally $\mu$-negligible set is locally $\nu$-negligible. 3) Every $\mu$-negligible compact set is $\nu$-negligible.

It is clear that 1) implies 2) (Cor. 1 of Prop. 3), and that 2) implies 3). We are going to show that 3) implies 1). We first note that if the condition 3) is satisfied, then every set $A$ that is universally measurable and locally $\mu$-negligible is locally $\nu$-negligible; for, $\nu^\bullet(A) = \sup \nu(K)$, where $K$ runs over the set of compact sets contained in $A$ (§ 1, No. 3, Prop. 10, a) and Ch. IV, § 4, No. 6, Cor. 2 of Th. 4). Next, we shall establish two lemmas.