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signed measure (Definition)

A signed measure on a measurable space $ (\Omega,\mathscr{S})$ is a function $ \mu:\mathscr{S}\rightarrow \mathbb{R}\cup\{+\infty\}$ which is $\sigma$ -additive and such that $\mu(\emptyset)=0$ .

Remarks.

  1. The usual (positive) measure is a particular case of signed measure, in which $|\mu| = \mu$ (see Jordan decomposition.)
  2. Notice that the value $-\infty$ is not allowed. For some authors, a signed measure can only take finite values (so that $+\infty$ 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 $\|\mu\| = |\mu|(\Omega)$ .
  3. An important example of signed measures arises from the usual measures in the following way: Let $ (\Omega,\mathscr{S},\mu)$ be a measure space, and let $f$ be a (real valued) measurable function such that $$\int_{\{x\in \Omega:f(x)<0\}} |f| d\mu <\infty.$$ Then a signed measure is defined by $$A\mapsto \int_A fd\mu.$$




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Cross-references: measurable function, real, measure space, norm, operations, normed vector space, finite, Jordan decomposition, measure, positive, function, measurable space
There are 12 references to this entry.

This is version 5 of signed measure, born on 2003-02-10, modified 2005-02-25.
Object id is 4013, canonical name is SignedMeasure.
Accessed 6524 times total.

Classification:
AMS MSC28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities)

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