[[Measure theory MOC]]
# Dominating measure

Given two measures $\mu,\nu$ on a [[Measure space|measurable space]] $(X,\Sigma)$ we say **$\mu$ dominates $\nu$** or **$\nu$ is absolutely continuous with respect to $\nu$**, written $\nu\ll\mu$, iff the following holds: #m/def/measure 

$$
\begin{align*}
\nu\ll\mu \iff (\forall A \in \Sigma)[\mu(A) = 0 \implies \nu(A)=0]
\end{align*}
$$ 
^eq

## Properties

- [[Radon-Nikodym theorem]]

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#state/tidy | #lang/en | #SemBr