signed measure
A signed measure on a measurable space is a function which is -additive (http://planetmath.org/Additive) and such that .
Remarks.
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1.
The usual (positive) measure is a particular case of signed measure, in which (see Jordan decomposition.)
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2.
Notice that the value is not allowed. For some authors, a signed measure can only take finite values (so that is not allowed either). This is sometimes useful because it turns the space of all signed measures into a normed vector space, with the natural operations, and the norm given by .
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3.
An important example of signed measures arises from the usual measures in the following way: Let be a measure space, and let be a (real valued) measurable function such that
Then a signed measure is defined by
Title | signed measure |
---|---|
Canonical name | SignedMeasure |
Date of creation | 2013-03-22 13:26:55 |
Last modified on | 2013-03-22 13:26:55 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 8 |
Author | Koro (127) |
Entry type | Definition |
Classification | msc 28A12 |