Jordan decomposition
Let be a signed measure space, and let be a Hahn decomposition for . We define and by
This definition is easily shown to be independent of the chosen Hahn decomposition.
It is clear that is a positive measure, and it is called the positive variation of . On the other hand, is a positive finite measure, called the negative variation of . The measure is called the total variation of .
Notice that . This decomposition of into its positive and negative parts is called the Jordan decomposition of .
Title | Jordan decomposition |
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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 |