Hahn Sets 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 set. A set $P\in\mathcal{F}$ is a Hahn set for $\nu$ if $\nu(A)\ge0$ for every $A\in\mathcal{F}$ with $A\subseteq P$, and $\nu(A)\le0$ for every $A\in\mathcal{F}$ with $A\subseteq X\setminus P$.