The total variation |nu|(E) from countable measurable partitions

Let $\nu$ be a signed measure or a complex measure on $(X,\mathcal{A})$. For $E\in \mathcal{A}$, define its total variation on $E$ by $\mathrm{\mid }\nu \mathrm{\mid }(E):=\sup \{{\sum }_{n=0}^{\mathrm{\infty }}\mathrm{\mid }\nu (E_n)\mathrm{\mid }:(E_n{)}_{n\in \mathbb{N}}\text{ is a countable measurable partition of }E\},$ where, for a signed measure, the term $\mathrm{\mid }\nu (E_n)\mathrm{\mid }$ means the ordinary absolute value when $\nu (E_n)\in \mathbb{R}$ and means $+\mathrm{\infty }$ when $\nu (E_n)\in \{+\mathrm{\infty },-\mathrm{\infty }\}$; for a complex measure it is the usual complex modulus. A countable measurable partition of $E$ means:

1. each $E_n$ lies in $\mathcal{A}$;
2. the sets $E_n$ are pairwise disjoint;
3. $E={\bigcup }_{n\in \mathbb{N}}{E}_n$.

The sum on the right is a nonnegative extended series, so it is always defined in $[0,+\mathrm{\infty }]\subseteq \bar{\mathbb{R}}$.

For signed measures, a later proposition shows that finite partitions already suffice. For complex measures, that finite-partition shortcut is not built into the definition here.