Existence of the Total Variation of a Finite Signed Measure

In the setting of Radon Measures on Locally Compact Metric Spaces: Standing Notation, let $(X,\mathcal{F})$ be a measurable space, let $\nu$ be a finite signed measure on $(X,\mathcal{F})$, and let $A\in\mathcal{F}$.

(i) Greatest element. The set of real numbers

$$S_{\nu}(A)=\{\nu(B)-\nu(A\setminus B):B\in\mathcal{F},\ B\subseteq A\}$$

has a greatest element, and for every Hahn set $P$ for $\nu$ this greatest element is

$$\max S_{\nu}(A)=\nu(A\cap P)-\nu(A\setminus P)=\nu^{+}(A)+\nu^{-}(A).$$