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Jordan Measure Decomposition


If mu 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

 mu^+=1/2(|mu|+mu),
(1)

where |mu| is the total variation. Similarly, the negative variation is

 mu^-=1/2(|mu|-mu).
(2)

Then the Jordan decomposition of mu is defined as

 mu=mu^+-mu^-.
(3)

When mu already is a positive measure then mu=mu^+. More generally, if mu is absolutely continuous, i.e.,

 mu(E)=int_Efdx,
(4)

then so are mu^+ and mu^-. The positive and negative variations can also be written as

 mu^+(E)=int_Ef^+dx
(5)

and

 mu^-(E)=int_Ef^-dx,
(6)

where f=f^+-f^- is the decomposition of f into its positive and negative parts.

The Jordan decomposition has a so-called minimum property. In particular, given any positive measure lambda, the measure mu has another decomposition

 mu=(mu^++lambda)-(mu^-+lambda).
(7)

The Jordan decomposition is minimal with respect to these changes. One way to say this is that any decomposition mu=lambda_1-lambda_2 must have lambda_1>=mu^+ and lambda_2>=mu^-.


See also

Measure, Polar Representation, Total Variation

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Jordan Measure Decomposition." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/JordanMeasureDecomposition.html

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