Banach-valued vector measure and variation

Let $(\mathrm{\Omega },\mathcal{A})$ be a measurable space and $X$ a real or complex Banach space. An $X$-valued vector measure is a map $\nu :\mathcal{A}\to X$ such that $\nu (\mathrm{\varnothing })=0$ and, for every pairwise disjoint sequence $(E_n)$ in $\mathcal{A}$,

$\nu \left({\bigcup }_{n=1}^{\mathrm{\infty }}{E}_n\right)={\sum }_{n=1}^{\mathrm{\infty }}\nu (E_n),$

where the series converges in the norm of $X$.

Its variation is the extended nonnegative set function

$\mathrm{\mid }\nu \mathrm{\mid }(E):=\sup \{{\sum }_{j=1}^m\mathrm{\parallel }\nu (E_j)\mathrm{\parallel }:(E_j{)}_{j=1}^m\text{ is a finite measurable partition of }E\}\mathrm{.}$

We set $\mathrm{\mid }\nu \mathrm{\mid }(\mathrm{\varnothing })=0$. The vector measure has bounded variation if $\mathrm{\mid }\nu \mathrm{\mid }(\mathrm{\Omega })<\mathrm{\infty }$.

Given a positive measure $\mu$ on $\mathcal{A}$, write $\nu \ll \mu$ when $\mu (E)=0$ implies $\nu (E)=0$ for every $E\in \mathcal{A}$.

Norm countable additivity, finiteness of variation, and absolute continuity with respect to a named scalar measure are three separate conditions. The one-cell partition gives $\mathrm{\parallel }\nu (E)\mathrm{\parallel }\le \mathrm{\mid }\nu \mathrm{\mid }(E)$, while the empty partition convention gives zero variation on the empty set.