Hahn and Jordan Decompositions of a Finite Signed Measure

In the setting of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, let $(X,\mathcal{F})$ be a measurable space and let $\nu$ be a finite signed measure on $(X,\mathcal{F})$.

(i) Hahn decomposition. Assume the axiom of dependent choice. Then $\nu$ has a Hahn set.

(ii) Jordan decomposition. Let $P$ be a Hahn set for $\nu$. Then the maps $\nu^{+}$ and $\nu^{-}$ on $\mathcal{F}$,

$$\nu^{+}(A)=\nu(A\cap P),\qquad\nu^{-}(A)=-\nu(A\setminus P),$$

take values in $[0,\infty]$ and are finite measures on $(X,\mathcal{F})$, and

$$\nu(A)=\nu^{+}(A)-\nu^{-}(A)\qquad(A\in\mathcal{F}).$$

They do not depend on the choice of $P$. If $\mu$ is a measure on $(X,\mathcal{F})$ and $\nu$ is absolutely continuous with respect to $\mu$, then $\nu^{+}$ and $\nu^{-}$ are absolutely continuous with respect to $\mu$.

(iii) Uniqueness. Let $P$ and $P'$ be Hahn sets for $\nu$. Then $\nu(A)=0$ for every $A\in\mathcal{F}$ with $A\subseteq(P\setminus P')\cup(P'\setminus P)$, and

$$\nu(A\cap P')=\nu(A\cap P),\qquad\nu(A\setminus P')=\nu(A\setminus P)\qquad(A\in\mathcal{F}).$$