Total Variation and Total Variation Norm 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 and let $\nu$ be a finite signed measure on $(X,\mathcal{F})$.

(i) Total variation. The total variation of $\nu$ is the function $|\nu|:\mathcal{F}\to\mathbb{R}$ with

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

the greatest element given by Existence of the Total Variation of a Finite Signed Measure (i).

(ii) Total variation norm. The total variation norm of $\nu$ is the real number

$$\|\nu\|=|\nu|(X).$$