# Zero Measure is Absolutely Continuous with respect to Every Measure

## Definition

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

Let $\mu$ be a measure on $\struct {M, \Sigma}$.

Let $\nu$ be the null measure on $\struct {M, \Sigma}$.

Then $\nu$ is absolutely continuous with respect to $\mu$.

## Proof

Let $E \in \Sigma$ be such that $\map \mu E = 0$.

By the definition of the null measure, we have:

$\map \nu E = 0$.

So whenever $E \in \Sigma$ is such that $\map \mu E = 0$, we have $\map \nu E = 0$.

So $\nu$ is absolutely continuous with respect to $\mu$.

$\blacksquare$