Vector measure

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

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

(*)

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.

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.

• ${\displaystyle \mu ,}$  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.

Given a vector measure ${\displaystyle \mu :{\mathcal {F}}\to X,}$  the variation ${\displaystyle |\mu |}$  of ${\displaystyle \mu }$  is defined as $${\displaystyle |\mu |(A)=\sup \sum _{i=1}^{n}\|\mu (A_{i})\|}$$  where the supremum is taken over all the partitions $${\displaystyle A=\bigcup _{i=1}^{n}A_{i}}$$  of ${\displaystyle A}$  into a finite number of disjoint sets, for all ${\displaystyle A}$  in ${\displaystyle {\mathcal {F}}.}$  Here, ${\displaystyle \|\cdot \|}$  is the norm on ${\displaystyle X.}$ 

The variation of ${\displaystyle \mu }$  is a finitely additive function taking values in ${\displaystyle [0,\infty ].}$  It holds that $${\displaystyle \|\mu (A)\|\leq |\mu |(A)}$$  for any ${\displaystyle A}$  in ${\displaystyle {\mathcal {F}}.}$  If ${\displaystyle |\mu |(\Omega )}$  is finite, the measure ${\displaystyle \mu }$  is said to be of bounded variation. One can prove that if ${\displaystyle \mu }$  is a vector measure of bounded variation, then ${\displaystyle \mu }$  is countably additive if and only if ${\displaystyle |\mu |}$  is countably additive.

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]

• Bochner measurable function
• Bochner integral – generalization of the Lebesgue integral to Banach-space valued functionsPages displaying wikidata descriptions as a fallback
• Bochner space – Type of topological space
• Complex measure – Measure with complex values
• Signed measure – Generalized notion of measure in mathematics
• Vector-valued functions – Function valued in a vector space; typically a real or complex onePages displaying short descriptions of redirect targets
• Weakly measurable function

• Cohn, Donald L. (1997) [1980]. Measure theory (reprint ed.). Boston–Basel–Stuttgart: Birkhäuser Verlag. pp. IX+373. ISBN 3-7643-3003-1. Zbl 0436.28001.
• Diestel, Joe; Uhl, Jerry J. Jr. (1977). Vector measures. Mathematical Surveys. Vol. 15. Providence, R.I: American Mathematical Society. pp. xiii+322. ISBN 0-8218-1515-6.
• Kluvánek, I., Knowles, G, Vector Measures and Control Systems, North-Holland Mathematics Studies 20, Amsterdam, 1976.
• van Dulst, D. (2001) [1994], "Vector measures", Encyclopedia of Mathematics, EMS Press
• Rudin, W (1973). Functional analysis. New York: McGraw-Hill. p. 114. ISBN 9780070542259.