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# Jordan decomposition

Let $(\Omega,\mathscr{S},\mu)$ be a signed measure space, and let $(A,B)$ be a Hahn decomposition for $\mu$. We define $\mu^{+}$ and $\mu^{-}$ by

$\mu^{+}(E)=\mu(A\cap E)\quad\mbox{and}\quad\mu^{-}(E)=-\mu(B\cap E).$ |

This definition is easily shown to be independent of the chosen Hahn decomposition.

It is clear that $\mu^{+}$ is a positive measure, and it is called the *positive variation* of $\mu$. On the other hand, $\mu^{-}$ is a positive finite measure, called the *negative variation* of $\mu$.
The measure $|\mu|=\mu^{+}+\mu^{-}$ is called the *total variation* of $\mu$.

Notice that $\mu=\mu^{+}-\mu^{-}$. This decomposition of $\mu$ into its positive and negative parts is called the *Jordan decomposition* of $\mu$.

Defines:

positive variation, negative variation, total variation

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

28A12*no label found*

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