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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 [[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>\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>\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

: <math>\lim_{n\to\infty}\left\|\mu\left(\displaystyle\bigcup_{i=n}^\infty A_i\right)\right\|=0, \quad\quad\quad (*)</math>

where <math>\|\cdot\|</math> is the norm on <math>X.</math>

Countably additive vector measures defined on [[sigma-algebra]]s are more general than [[measure (mathematics)|measures]], [[signed measure]]s, and [[complex measure]]s, which are [[countably additive function]]s taking values respectively on the [[extended real line|extended 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>\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, we get two different outcomes.

* <math>\mu,</math> viewed as a function from <math>\mathcal F</math> to the [[Lp space|''L''''p''-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 ''L''''p''-space <math>L^1([0, 1]),</math> 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 <math>\mu:\mathcal{F}\to X,</math> the '''variation''' <math>|\mu|</math> of <math>\mu</math> is defined as

: <math>|\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>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>||\mu(A)||\le |\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]]) vector 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 '''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
| pages =
| 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=harv | postscript={{inconsistent citations}}}}</ref>  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|authorlink=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|accessdate=7 February 2011|series=Research paper|volume=892|publisher=Graduate School of Business, Stanford University|location=Palo Alto, CA|id=(Draft of articles for the first edition of ''New Palgrave Dictionary of Economics'')|ref=harv|postscript={{inconsistent citations}}}}</ref><ref name="Aumann" >{{cite journal|authorlink=Robert Aumann|first=Robert 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}} This paper builds on two papers by Aumann: {{cite journal||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}} 
 {{cite journal||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}} </ref><ref>{{cite news|last=Vind|first=Karl|year=1964|title=Edgeworth-allocations in an exchange economy with many traders|journal=International Economic Review|volume=5|pages=165–77|number=2|month=May|ref=harv|jstor=2525560}} Vind's article was noted by {{harvtxt|Debreu|1991|p=4}} with this comment:
<blockquote>
The concept of a convex set (i.e., a set containing the segment connecting any two of its points) had repeatedly been placed at the center of economic theory before 1964. It appeared in a new light with the introduction of integration theory in the study of economic competition: If one associates with every agent of an economy an arbitrary set in the commodity space and ''if one averages those individual 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. A. Lyapunov, see {{harvtxt|Vind|1964}}."] But explanations of the  ... functions of prices  ... can be made to rest on the ''convexity of sets derived by that averaging process''. ''Convexity'' in the commodity space ''obtained by aggregation'' over a collection of insignificant agents is an insight that economic theory owes  ... to integration theory. [''Italics added'']
</blockquote>
{{cite news|title=The Mathematization of economic theory|first=Gérard|last=Debreu|authorlink=Gérard Debreu|issue=Presidential address delivered at the 103rd meeting of the American Economic Association, 29 December 1990, Washington, DC|journal=The American Economic Review |volume=81, number 1 |date=March 1991 |pages=1–7 |jstor=2006785}}
</ref> in ([[bang–bang control|"bang–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 P.|series=Mathematics in Science and Engineering|volume=56|publisher=Academic Press|location=New York—London|year=1969|pages=viii+136|mr=420366|unused_data=}}</ref><ref name="Artstein"/> and in [[statistical theory]].<ref name="Artstein" >{{cite news|last=Artstein|first=Zvi|title=Discrete and continuous bang-bang and facial 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 news|last=Tardella|first=Fabio|title=A new proof of the Lyapunov convexity 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 M.|authorlink=Ross Starr|chapter=Shapley–Folkman theorem|title=The New Palgrave Dictionary of Economics|editor-first=Steven N.|editor-last=Durlauf|editor2-first=Lawrence E., ed.|editor2-last=Blume|publisher=Palgrave Macmillan|year=2008|edition=Second|pages=317–318 (1st ed.)|url=http://www.dictionaryofeconomics.com/article?id=pde2008_S000107|doi=10.1057/9780230226203.1518|ref=harv}}</ref>
<ref>Page 210: {{cite news|last=Mas-Colell|first=Andreu|authorlink=Andreu Mas-Colell|title=A note on the core 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>

==References==
{{Reflist}}

===Books===
*{{Cite book
| last = Cohn
| first = Donald L.
| title = Measure theory
| place = Boston–Basel–Stuttgart
| publisher = [[Birkhäuser Verlag]]
| origyear = 1980
| year = 1997
| edition = reprint
| pages = IX+373
| url = http://books.google.it/books?id=vRxV2FwJvoAC&printsec=frontcover&dq=Measure+theory+Cohn&cd=1#v=onepage&q&f=false
| doi =
| zbl = 0436.28001
| isbn = 3-7643-3003-1
| ref = harv
| postscript = {{inconsistent citations}}
}}
*{{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 '''20''', Amsterdam, 1976.
* {{springerEOM|title=Vector measures|id=Vector_measure|first=D. |last=van Dulst}}

==See also==

*[[Bochner integral]]
{{Use dmy dates|date=September 2011}}

{{Functional Analysis}}

[[Category:Measures (measure theory)]]
[[Category:Functional analysis]]
[[Category:Control theory]]
[[Category:Mathematical and quantitative methods (economics)]]

Basal area - Revision history

en>ChrisGualtieri: Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB