Jordan decomposition

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

$${\mu }^{+}(E)=\mu (A\cap E)\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\mu }^{-}(E)=-\mu (B\cap E).$$

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 $\mu$.