# 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 $\mu$.

Title Jordan decomposition JordanDecomposition 2013-03-22 13:27:02 2013-03-22 13:27:02 Koro (127) Koro (127) 9 Koro (127) Definition msc 28A12 positive variation negative variation total variation