If is a real measure (i.e., a measure that takes on real values), then one can decompose it according to where it is positive and negative. The positive variation is defined by
(1)
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where is the total variation. Similarly, the negative variation is
(2)
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Then the Jordan decomposition of is defined as
(3)
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When already is a positive measure then . More generally, if is absolutely continuous, i.e.,
(4)
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then so are and . The positive and negative variations can also be written as
(5)
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and
(6)
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where is the decomposition of into its positive and negative parts.
The Jordan decomposition has a so-called minimum property. In particular, given any positive measure , the measure has another decomposition
(7)
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The Jordan decomposition is minimal with respect to these changes. One way to say this is that any decomposition must have and .