Category:Jordan Decompositions
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This category contains results about Jordan Decompositions.
Definitions specific to this category can be found in Definitions/Jordan Decompositions.
Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $\tuple {P, N}$ be a Hahn decomposition of $\mu$.
Define:
- $\map {\mu^+} A = \map \mu {A \cap P}$
and:
- $\map {\mu^-} A = -\map \mu {A \cap N}$
for each $A \in \Sigma$.
Then from the Jordan Decomposition Theorem, we have:
- $\mu = \mu^+ - \mu^-$
and we say that $\tuple {\mu^+, \mu^-}$ is the Jordan decomposition corresponding to $\tuple {P, N}$.
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