Definition:Jordan Decomposition

Definition
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 $\tuple {P, N}$.

Also known as
A Jordan decomposition is also known as a Hahn-Jordan decomposition (with ).

Some sources refer to it as a Hahn decomposition, but recognises the fact that this is a specific refinement of that concept for which it is worth having a separate term.

Also see

 * The Jordan Decomposition Theorem shows that $\mu^+$ and $\mu^-$ are well-defined, and are measures with $\mu = \mu^+ - \mu^-$.
 * Uniqueness of Jordan Decomposition shows that the Jordan decomposition of $\mu$ is independent of $\tuple {P, N}$, allowing us to refer to the Jordan decomposition of $\mu$ without qualification.