Category:Jordan Decompositions

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 $\tuple {P, N}$.