absolutely continuous

Let μ and ν be signed measures or complex measuresMathworldPlanetmath on the same measurable spaceMathworldPlanetmathPlanetmath (Ω,𝒮). We say that ν is absolutely continuousMathworldPlanetmath with respect to μ if, for each A𝒮 such that |μ|(A)=0, it holds that ν(A)=0. This is usually denoted by νμ.


If μ and ν are signed measures and (ν+,ν-) is the Jordan decomposition of ν, the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.


  2. 2.

    ν+μ and ν-μ;

  3. 3.


If ν is a finite signed or complex measure and νμ, the following useful property holds: for each ε>0, there is a δ>0 such that |ν|(E)<ε whenever |μ|(E)<δ.

Title absolutely continuous
Canonical name AbsolutelyContinuous
Date of creation 2013-03-22 13:26:12
Last modified on 2013-03-22 13:26:12
Owner Koro (127)
Last modified by Koro (127)
Numerical id 10
Author Koro (127)
Entry type Definition
Classification msc 28A12
Related topic RadonNikodymTheorem
Related topic AbsolutelyContinuousFunction2
Defines absolute continuity