# absolutely continuous

Let $\mu$ and $\nu$ be signed measures or complex measures on the same measurable space $(\Omega,\mathscr{S})$. We say that $\nu$ is absolutely continuous with respect to $\mu$ if, for each $A\in\mathscr{S}$ such that $|\mu|(A)=0$, it holds that $\nu(A)=0$. This is usually denoted by $\nu\ll\mu$.

Remarks.

If $\mu$ and $\nu$ are signed measures and $(\nu^{+},\nu^{-})$ is the Jordan decomposition of $\nu$, the following are equivalent:

1. 1.

$\nu\ll\mu$;

2. 2.

$\nu^{+}\ll\mu$ and $\nu^{-}\ll\mu$;

3. 3.

$|\nu|\ll|\mu|$.

If $\nu$ is a finite signed or complex measure and $\nu\ll\mu$, the following useful property holds: for each $\varepsilon>0$, there is a $\delta>0$ such that $|\nu|(E)<\varepsilon$ whenever $|\mu|(E)<\delta$.

Title absolutely continuous AbsolutelyContinuous 2013-03-22 13:26:12 2013-03-22 13:26:12 Koro (127) Koro (127) 10 Koro (127) Definition msc 28A12 RadonNikodymTheorem AbsolutelyContinuousFunction2 absolute continuity