Jordan decomposition

Let (Ω,𝒮,μ) be a signed measure space, and let (A,B) be a Hahn decomposition for μ. We define μ+ and μ- by


This definition is easily shown to be independent of the chosen Hahn decomposition.

It is clear that μ+ is a positive measureMathworldPlanetmath, and it is called the positive variation of μ. On the other hand, μ- is a positive finite measure, called the negative variation of μ. The measure |μ|=μ++μ- is called the total variationMathworldPlanetmath of μ.

Notice that μ=μ+-μ-. This decomposition of μ into its positive and negative parts is called the Jordan decomposition of μ.

Title Jordan decomposition
Canonical name JordanDecomposition
Date of creation 2013-03-22 13:27:02
Last modified on 2013-03-22 13:27:02
Owner Koro (127)
Last modified by Koro (127)
Numerical id 9
Author Koro (127)
Entry type Definition
Classification msc 28A12
Defines positive variation
Defines negative variation
Defines total variation