User:Caliburn/s/mt/Existence of Finite Measure Sharing Null Sets with Sigma-Finite Measure

Theorem

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

Let $\mu$ be a $\sigma$-finite measure on $\struct {X, \Sigma}$.

Then there exists a finite measure $\nu$ on $\struct {X, \Sigma}$ such that:

$A \in \Sigma$ is $\mu$-null if and only if it is $\nu$-null.