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Jordan decomposition (Definition)

Let $ (\Omega,\mathscr{S},\mu)$ be a signed measure space, and let $ (A,B)$ be a Hahn decomposition for $ \mu$. We define $ \mu^+$ and $ \mu^-$ by

$\displaystyle \mu^+(E) = \mu(A\cap E)$   and$\displaystyle \quad \mu^-(E)=-\mu(B\cap E).$
This definition is easily shown to be independent of the chosen Hahn decomposition.

It is clear that $ \mu^+$ is a positive measure, and it is called the positive variation of $ \mu$. On the other hand, $ \mu^-$ is a positive finite measure, called the negative variation of $ \mu$. The measure $ \vert\mu\vert=\mu^+ + \mu^-$ is called the total variation of $ \mu$.

Notice that $ \mu = \mu^+ - \mu^-$. This decomposition of $ \mu$ into its positive and negative parts is called the Jordan decomposition of $ \mu$.



"Jordan decomposition" is owned by Koro.
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Also defines:  positive variation, negative variation, total variation
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Cross-references: negative, finite, measure, positive, clear, independent, Hahn decomposition, signed measure
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This is version 6 of Jordan decomposition, born on 2003-02-10, modified 2003-02-11.
Object id is 4015, canonical name is JordanDecomposition.
Accessed 7852 times total.

Classification:
AMS MSC: 28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities)
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