Zero Measure is Absolutely Continuous with respect to Every Measure
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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 $\mu$.
$\blacksquare$
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): absolutely continuous: 2.