Theorem 16.1.8 (Jordan Decomposition). Let $(X, \cm)$ be a measurable space and $\mu: \cm \to [-\infty, \infty]$ be a signed measure on $X$, then there exists positive measures $\mu^{+}, \mu^{-}: \cm \to [0, \infty]$ such that
$\mu^{+} \perp \mu^{-}$.
$\mu = \mu^{+} - \mu^{-}$.
For any other pair $(\nu^{+}, \nu^{-})$ satisfying (1) and (2), $\mu^{+} = \nu^{+}$ and $\mu^{-} = \nu^{-}$.
Proof. (1), (2): Let $X = P \sqcup N$ be a Hahn decomposition of $\mu$. For each $E \in \cm$, let
then $(\mu^{+}, \mu^{-})$ satisfies (1) and (2).
(U): Let $X = P' \sqcup N'$ such that $P'$ is $\nu^{-}$-null and $N'$ is $\nu^{+}$-null, then $X = P' \sqcup N'$ is a Hahn decomposition of $\mu$. Since $P' \Delta P = 0$ and $N' \Delta N = 0$, for any $E \in \cm$,
Likewise,