Total variation of a measure

Summary

Let $\nu$ be a signed measure or a complex measure. Its total variation $\abs{\nu}$ is the smallest positive measure with $\abs{\nu(E)} \le \abs{\nu}(E)$ for every set $E$. It is computed as the supremum of $\sum \abs{\nu(E_i)}$ over partitions of a set into measurable pieces. For a signed measure it equals $\nu^+ + \nu^-$ from the Jordan decomposition. The number $\abs{\nu}(X)$ is the total variation norm, which makes the finite signed or complex measures a Banach space. It extends the total variation of a function to measures.