In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only.

Given a field of sets ${\displaystyle (\Omega ,{\mathcal {F}})}$ and a Banach space ${\displaystyle X,}$ a finitely additive vector measure (or measure, for short) is a function ${\displaystyle \mu :{\mathcal {F}}\to X}$ such that for any two disjoint sets ${\displaystyle A}$ and ${\displaystyle B}$ in ${\displaystyle {\mathcal {F}}}$ one has $${\displaystyle \mu (A\cup B)=\mu (A)+\mu (B).}$$

A vector measure ${\displaystyle \mu }$ is called countably additive if for any sequence ${\displaystyle (A_{i})_{i=1}^{\infty }}$ of disjoint sets in ${\displaystyle {\mathcal {F}}}$ such that their union is in ${\displaystyle {\mathcal {F}}}$ it holds that $${\displaystyle \mu {\left(\bigcup _{i=1}^{\infty }A_{i}\right)}=\sum _{i=1}^{\infty }\mu (A_{i})}$$ with the series on the right-hand side convergent in the norm of the Banach space ${\displaystyle X.}$

It can be proved that an additive vector measure ${\displaystyle \mu }$ is countably additive if and only if for any sequence ${\displaystyle (A_{i})_{i=1}^{\infty }}$ as above one has

where ${\displaystyle \|\cdot \|}$ is the norm on ${\displaystyle X.}$

Countably additive vector measures defined on sigma-algebras are more general than finite measures, finite signed measures, and complex measures, which are countably additive functions taking values respectively on the real interval ${\displaystyle [0,\infty ),}$ the set of real numbers, and the set of complex numbers.

Consider the field of sets made up of the interval ${\displaystyle [0,1]}$ together with the family ${\displaystyle {\mathcal {F}}}$ of all Lebesgue measurable sets contained in this interval. For any such set ${\displaystyle A,}$ define $${\displaystyle \mu (A)=\chi _{A}}$$ where ${\displaystyle \chi }$ is the indicator function of ${\displaystyle A.}$ Depending on where ${\displaystyle \mu }$ is declared to take values, two different outcomes are observed.

Both of these statements follow quite easily from the criterion (*) stated above.