Characterization of Measures that Share Null Sets

Theorem
Let $\struct {X, \Sigma}$ be a measurable space.

Let $\mu$ and $\nu$ be $\sigma$-finite measures on $\struct {X, \Sigma}$.


 * $(1) \quad$ $\nu \ll \mu$ and $\mu \ll \nu$, where $\ll$ denotes absolutely continuity
 * $(2) \quad$ $A \in \Sigma$ is a $\mu$-null set it is a $\nu$-null set
 * $(3) \quad$ there exists a positive $\Sigma$-measurable function $g : X \to \R$ such that:
 * $\ds \map \nu A = \int_A g \rd \mu$
 * for each $A \in \Sigma$.