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{{Use dmy dates|date=April 2022}}
In [[mathematics]], a '''vector measure''' is a [[function (mathematics)|function]] defined on a [[family of sets]] and taking [[vector space|vector]] values satisfying certain properties. It is a generalization of the concept of finite [[measure (mathematics)|measure]], which takes [[nonnegative]] [[real number|real]] values only.

== Definitions and first consequences ==

Given a [[field of sets]] <math>(\Omega, \mathcal F)</math> and a [[Banach space]] <math>X,</math> a '''finitely additive vector measure''' (or '''measure''', for short) is a function <math>\mu:\mathcal {F} \to X</math> such that for any two [[disjoint set]]s <math>A</math> and <math>B</math> in <math>\mathcal{F}</math> one has
<math display="block">\mu(A\cup B) =\mu(A) + \mu (B).</math>

A vector measure <math>\mu</math> is called '''countably additive''' if for any [[sequence]] <math>(A_i)_{i=1}^{\infty}</math> of disjoint sets in <math>\mathcal F</math> such that their union is in <math>\mathcal F</math> it holds that
<math display="block">\mu{\left(\bigcup_{i=1}^\infty A_i\right)} = \sum_{i=1}^{\infty}\mu(A_i)</math>
with the [[series (mathematics)|series]] on the right-hand side convergent in the [[norm (mathematics)|norm]] of the Banach space <math>X.</math>

It can be proved that an additive vector measure <math>\mu</math>  is countably additive if and only if for any sequence <math>(A_i)_{i=1}^{\infty}</math> as above one has
{{NumBlk||<math display="block">\lim_{n\to\infty} \left\|\mu{\left(\bigcup_{i=n}^\infty A_i\right)}\right\| = 0, </math>|{{EquationRef|<nowiki>*</nowiki>}}}}
where <math>\|\cdot\|</math> is the norm on <math>X.</math>

Countably additive vector measures defined on [[sigma-algebra]]s are more general than finite [[measure (mathematics)|measure]]s, finite [[signed measure]]s, and [[complex measure]]s, which are [[countably additive function]]s taking values respectively on the real interval <math>[0, \infty),</math> the set of [[real number]]s, and the set of [[complex number]]s.

== Examples ==

Consider the field of sets made up of the interval <math>[0, 1]</math> together with the family <math>\mathcal F</math> of all [[Lebesgue measurable set]]s contained in this interval. For any such set <math>A,</math> define
<math display="block">\mu(A) = \chi_A</math>
where <math>\chi</math> is the [[indicator function]] of <math>A.</math> Depending on where <math>\mu</math> is declared to take values, two different outcomes are observed.

* <math>\mu,</math> viewed as a function from <math>\mathcal F</math> to the [[Lp space|<math>L^p</math>-space]] <math>L^\infty([0, 1]),</math> is a vector measure which is not countably-additive.
* <math>\mu,</math> viewed as a function from <math>\mathcal F</math> to the <math>L^p</math>-space <math>L^1([0, 1]),</math> is a countably-additive vector measure.

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

== The variation of a vector measure==

Given a vector measure <math>\mu : \mathcal{F} \to X,</math> the '''variation'''  <math>|\mu|</math> of <math>\mu</math> is defined as
<math display="block">|\mu|(A)=\sup \sum_{i=1}^n \|\mu(A_i)\|</math>
where the [[supremum]] is taken over all the [[Partition of a set|partitions]]
<math display="block">A = \bigcup_{i=1}^n A_i</math>
of <math>A</math> into a finite number of disjoint sets, for all <math>A</math> in <math>\mathcal{F}.</math> Here, <math>\|\cdot\|</math> is the norm on <math>X.</math>

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

== Lyapunov's theorem ==

In the theory of vector measures, ''[[Alexey Lyapunov|Lyapunov]]<nowiki>'s theorem</nowiki>'' states that the range of a ([[atom (measure theory)|non-atomic]]) finite-dimensional vector&nbsp;measure is [[closed set|closed]] and [[convex set|convex]].<ref name="KluvanekKnowles">[[Igor Kluvánek|Kluvánek, I.]], Knowles, G., ''Vector Measures and Control Systems'', North-Holland Mathematics Studies&nbsp;'''20''', Amsterdam, 1976.</ref><ref name="DiestelUhl" >{{cite book|last1=Diestel|first1=Joe| last2=Uhl|first2=Jerry J. Jr.|title=Vector measures|publisher=American Mathematical Society|location=Providence, R.I|year=1977|isbn=0-8218-1515-6}}</ref><ref name="RolewiczControl">{{Cite book|title=Functional analysis and control theory: Linear systems|last=Rolewicz |first=Stefan|year=1987| isbn=90-277-2186-6| publisher=D. Reidel Publishing Co.; PWN—Polish Scientific Publishers|oclc=13064804|edition=Translated from the Polish by Ewa Bednarczuk|series=Mathematics and its Applications (East European Series)|location=Dordrecht; Warsaw|volume=29|pages=xvi+524|mr=920371}}</ref> <!-- Lyapunov's theorem --> 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 [[zonotope]]s).<ref name="DiestelUhl"/> It is used in [[mathematical economics|economics]],<ref>{{Cite book | last=Roberts|first=John |author-link=Donald John Roberts|chapter=Large economies|title=Contributions to the ''New Palgrave'' |editor=David M. Kreps|editor1-link=David M. Kreps|editor2=John Roberts|editor2-link=Donald John Roberts|editor3=Robert B. Wilson |editor3-link=Robert B. Wilson|date=July 1986|pages=30–35|url=https://gsbapps.stanford.edu/researchpapers/library/RP892.pdf |access-date=7 February 2011|series=Research paper|volume=892|publisher=Graduate School of Business, Stanford University |location=Palo Alto,&nbsp;CA|id=(Draft of articles for the first edition of ''New&nbsp;Palgrave Dictionary of Economics'')}}</ref><ref name="Aumann" >{{cite journal|author-link=Robert Aumann|first=Robert&nbsp;J.|last=Aumann|title=Existence of competitive equilibrium in markets with a continuum of traders|journal=Econometrica|volume=34|number=1|date=January 1966 |pages=1–17 |jstor=1909854|mr=191623|doi=10.2307/1909854|s2cid=155044347 }} This paper builds on two papers by Aumann: <p> {{cite journal <!-- | authorlink=Robert Aumann-->|title=Markets with a continuum of traders |journal=Econometrica |volume=32|number=1–2|date=January–April 1964|pages=39–50|jstor=1913732|mr=172689 |doi=10.2307/1913732 |last1=Aumann |first1=Robert J.}}</p>
<p> {{cite journal <!-- authorlink=Robert Aumann|first=Robert&nbsp;J.|last=Aumann -->|title=Integrals of set-valued functions |journal=Journal of Mathematical Analysis and Applications|volume=12|number=1|date=August 1965|pages=1–12 | doi=10.1016/0022-247X(65)90049-1 |mr=185073|last1=Aumann|first1=Robert J.}}</p></ref><ref>{{cite news|last=Vind|first=Karl | date=May 1964|title=Edgeworth-allocations in an exchange economy with many traders|journal=International Economic Review | volume=5|pages=165–77|number=2|jstor=2525560}} Vind's article was noted by {{harvtxt|Debreu|1991|p=4}} with this comment:
<blockquote>
The concept of a convex&nbsp;set (i.e., a set containing the segment connecting any two of its points) had repeatedly been placed at the center of economic&nbsp;theory before&nbsp;1964. It appeared in a new&nbsp;light with the introduction of integration&nbsp;theory in the study of economic&nbsp;competition: If<!-- original "if" inconsistent with our capitization  --> one associates with every&nbsp;agent of an economy an arbitrary&nbsp;set in the commodity&nbsp;space and ''if one averages those individual&nbsp;sets'' over a collection of insignificant agents, ''then the resulting set is necessarily convex''. [Debreu appends this footnote: "On this direct consequence of a theorem of A.&nbsp;A.&nbsp;Lyapunov, see {{harvtxt|Vind|1964}}."] But explanations of the <!-- three --> ... functions of prices <!-- taken as examples --> ... can be made to rest&nbsp;on the ''convexity of sets derived by that averaging&nbsp;process''. ''Convexity'' in the commodity&nbsp;space ''obtained by aggregation'' over a collection of insignificant&nbsp;agents is an insight that economic&nbsp;theory owes <!-- in its revealing clarity --> ... to integration&nbsp;theory. [''Italics added'']
</blockquote>
{{cite news|title=The Mathematization of economic theory|first=Gérard|last=Debreu|author-link=Gérard Debreu|issue=Presidential address delivered at the&nbsp;103rd meeting of the American Economic Association,&nbsp;29 December&nbsp;1990, Washington,&nbsp;DC | journal=The American Economic Review |volume=81, number 1|date=March 1991|pages=1–7|jstor=2006785}}
</ref> in ([[bang–bang control|"bang&ndash;bang"]]) [[control theory]],<ref name="KluvanekKnowles"/><ref name="RolewiczControl"/><ref>{{cite book|title=Functional analysis and time optimal control|last1=Hermes|first1=Henry |last2=LaSalle|first2=Joseph&nbsp;P.|series=Mathematics in Science and Engineering|volume=56|publisher=Academic Press|location=New York—London|year=1969|pages=viii+136|mr=420366}}</ref><ref name="Artstein"/> and in [[statistical theory]].<ref name="Artstein" >{{cite journal|last=Artstein|first=Zvi|title=Discrete&nbsp;and&nbsp;continuous bang-bang and facial&nbsp;spaces, or: Look for the extreme points|journal=SIAM Review|volume=22|year=1980|number=2|pages=172–185|doi=10.1137/1022026|jstor=2029960|mr=564562}}</ref>
Lyapunov's theorem has been proved by using the [[Shapley–Folkman lemma]],<ref>{{cite journal|last=Tardella|first=Fabio|title=A new proof of the Lyapunov convexity&nbsp;theorem|journal=SIAM Journal on Control and Optimization|volume=28|year=1990|number=2| pages=478–481|doi=10.1137/0328026|mr=1040471}}</ref> which has been viewed as a [[discretization|discrete]] [[discrete mathematics#Discrete analogues of continuous mathematics|analogue]] of Lyapunov's theorem.<ref name="Artstein" /><ref name="Starr08" >{{cite book|last=Starr|first=Ross&nbsp;M.|author-link=Ross Starr|chapter=Shapley–Folkman theorem|title=The New&nbsp;Palgrave Dictionary of Economics|editor-first=Steven&nbsp;N.|editor-last=Durlauf|editor2-first=Lawrence&nbsp;E. |editor2-last=Blume|publisher=Palgrave Macmillan|year=2008|edition=Second|pages=317–318 | url=http://www.dictionaryofeconomics.com/article?id=pde2008_S000107|doi=10.1057/9780230226203.1518| isbn=978-0-333-78676-5}}</ref><ref>Page 210: {{cite journal|last=Mas-Colell|first=Andreu|author-link=Andreu Mas-Colell|title=A note on the core&nbsp;equivalence theorem: How many blocking coalitions are there?|journal=Journal of Mathematical Economics| volume=5| year=1978| number=3| pages=207–215|doi=10.1016/0304-4068(78)90010-1|mr=514468}}</ref>

== See also ==

* {{annotated link|Bochner measurable function}}
* {{annotated link|Bochner integral}}
* {{annotated link|Bochner space}}
* {{annotated link|Complex measure}}
* {{annotated link|Signed measure}}
* {{annotated link|Vector-valued functions}}
* {{annotated link|Weakly measurable function}}

== References ==

{{reflist|group=note}}
{{reflist}}

== Bibliography ==

* {{cite book|last=Cohn|first=Donald L.|title=Measure theory|place=Boston&ndash;Basel&ndash;Stuttgart|publisher=[[Birkhäuser Verlag]]|orig-year=1980|year=1997|edition=reprint|pages=IX+373|url=https://books.google.com/books?id=vRxV2FwJvoAC&q=Measure+theory+Cohn|zbl=0436.28001|isbn=3-7643-3003-1}}
*{{cite book|last1 =Diestel|first1=Joe|last2 =Uhl|first2=Jerry J. Jr.|title =Vector measures|series=Mathematical Surveys|volume=15|publisher=American Mathematical Society|location=Providence, R.I|year=1977|pages=xiii+322|isbn=0-8218-1515-6}}
* [[Igor Kluvánek|Kluvánek, I.]], Knowles, G, ''Vector Measures and Control Systems'', North-Holland Mathematics Studies&nbsp;'''20''', Amsterdam, 1976.
* {{springerEOM|title=Vector measures|id=Vector_measure|first=D. |last=van Dulst}}
* {{cite book|last1=Rudin|first1=W|title=Functional analysis|url=https://archive.org/details/functionalanalys00rudi_320|url-access=limited|date=1973|publisher=McGraw-Hill|location=New York|page=[https://archive.org/details/functionalanalys00rudi_320/page/n123 114]|isbn=9780070542259}}

{{Analysis in topological vector spaces}}
{{Functional analysis}}
{{Measure theory}}

[[Category:Control theory]]
[[Category:Functional analysis]]
[[Category:Measures (measure theory)]]