Definition:Jordan Decomposition

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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 $\mu$.


$\map {\mu^+} A = \map \mu {A \cap P}$


$\map {\mu^-} A = -\map \mu {A \cap N}$

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 Hans Hahn).

Some sources refer to it as a Hahn decomposition, but $\mathsf{Pr} \infty \mathsf{fWiki}$ recognises the fact that this is a specific refinement of that concept for which it is worth having a separate term.

Also see

Source of Name

This entry was named for Marie Ennemond Camille Jordan.