Zero Measure is Absolutely Continuous with respect to Every Measure

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Definition

Let $M$ be a measurable space.

Let $\mu$ be a measure on $M$.

Let $\nu$ be the null measure on $M$.

Let $\nu$ whenever $\map {\mu_2} E = 0$.


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


Proof

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


By definition of null measure:

$\map \nu E = 0$

from which the result follows trivially.

$\blacksquare$


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