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.

Definitions and first consequences

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

$${\displaystyle \mu {\left(\bigcup _{i=1}^{\infty }A_{i}\right)}=\sum _{i=1}^{\infty }\mu (A_{i})}$$

${\displaystyle X.}$

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

$${\displaystyle \lim _{n\to \infty }\left\|\mu {\left(\bigcup _{i=n}^{\infty }A_{i}\right)}\right\|=0,}$$

(*)

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.

Examples

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

$${\displaystyle \mu (A)=\chi _{A))$$

${\displaystyle \chi }$

${\displaystyle A.}$

${\displaystyle \mu }$

• ${\displaystyle \mu ,}$ viewed as a function from ${\displaystyle {\mathcal {F))}$ to the ${\displaystyle L^{p))$-space ${\displaystyle L^{\infty }([0,1]),}$ is a vector measure which is not countably-additive.
• ${\displaystyle \mu ,}$ viewed as a function from ${\displaystyle {\mathcal {F))}$ to the ${\displaystyle L^{p))$-space ${\displaystyle L^{1}([0,1]),}$ is a countably-additive vector measure.

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

The variation of a vector measure

Given a vector measure ${\displaystyle \mu :{\mathcal {F))\to X,}$ the variation ${\displaystyle |\mu |}$ of ${\displaystyle \mu }$ is defined as

$${\displaystyle |\mu |(A)=\sup \sum _{i=1}^{n}\|\mu (A_{i})\|}$$

$${\displaystyle A=\bigcup _{i=1}^{n}A_{i))$$

${\displaystyle A}$

${\displaystyle A}$

${\displaystyle {\mathcal {F)).}$

${\displaystyle \|\cdot \|}$

${\displaystyle X.}$

The variation of ${\displaystyle \mu }$ is a finitely additive function taking values in ${\displaystyle [0,\infty ].}$ It holds that

$${\displaystyle \|\mu (A)\|\leq |\mu |(A)}$$

${\displaystyle A}$

${\displaystyle {\mathcal {F)).}$

${\displaystyle |\mu |(\Omega )}$

${\displaystyle \mu }$

${\displaystyle \mu }$

${\displaystyle \mu }$

${\displaystyle |\mu |}$

Lyapunov's theorem

In the theory of vector measures, Lyapunov's theorem states that the range of a (non-atomic) finite-dimensional vector measure is closed and convex.^[1][2][3] In fact, the range of a non-atomic vector measure is a zonoid (the closed and convex set that is the limit of a convergent sequence of zonotopes).^[2] It is used in economics,^[4][5][6] in ("bang–bang") control theory,^[1][3][7][8] and in statistical theory.^[8] Lyapunov's theorem has been proved by using the Shapley–Folkman lemma,^[9] which has been viewed as a discrete analogue of Lyapunov's theorem.^[8][10][11]