查看“︁向量测度”︁的源代码

'''向量测度'''（vector measure）是[[数学]]名詞，是指針對[[集合族]]定義的函數，其值為滿足特定性質的[[向量]]。向量测度是[[测度]]概念的推廣，测度是針對[[集合 (數學)|集合]]定義的函數，函數的值只有非負的[[實數]]。
==定義及相關推論==
給定[[集合域]] <math>(\Omega, \mathcal F)</math>及[[巴拿赫空间]] <math>X</math>，'''有限加性向量測度'''（finitely additive vector measure）簡稱測度，是一個滿足以下條件的函數<math>\mu:\mathcal {F} \to X</math>：針對任二個<math>\mathcal{F}</math>內的[[不交集]]<math>A</math>和<math>B</math>，下式均成立：

: <math>\mu(A\cup B) =\mu(A) + \mu (B).</math>

向量测度<math>\mu</math>稱為'''可數加性'''（countably additive）若針對任意<math>\mathcal F</math>內[[不交集]]形成的[[序列]] <math>(A_i)_{i=1}^{\infty}</math>，都可以讓<math>\mathcal F</math>內的聯集滿足以下條件

: <math>\mu\left(\bigcup_{i=1}^\infty A_i\right) =\sum_{i=1}^{\infty}\mu(A_i)</math>

等號右邊的[[级数]]會收斂到巴拿赫空间<math>X</math>的[[范数]]。

可以證明向量測度<math>\mu</math>有可數加性，若且唯若針對任何以上的序列<math>(A_i)_{i=1}^{\infty}</math>，下式均成立

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

其中<math>\|\cdot\|</math>是<math>X</math>的範數。

在[[Σ-代数]]中定義的可數加性向量测度，會比有限[[测度]]（测度的值為非負數）、有限{{link-en|有號測度|signed measure}}（测度的值為實數）及{{link-en|複數测度|complex measure}}（测度的值為複數）要廣泛。

==舉例==
考慮一個由<math>[0, 1]</math>區間的集合形成的場，以及此區間內所有[[勒贝格测度]]形成的族<math>\mathcal F</math>。針對任意集合<math>A</math>，定義

: <math>\mu(A)=\chi_A</math>

其中<math>\chi</math>是<math>A</math>的[[指示函数]]。依<math>\mu</math>的定義不同，會得到不同的結果。

* <math>\mu</math>若是從<math>\mathcal F</math>到[[Lp空间]] <math>L^\infty([0, 1])</math>的函數，<math>\mu</math>是沒有可數加性的向量测度。
* <math>\mu</math>若是從<math>\mathcal F</math>到''L''<sup>''p''</sup>空間 <math>L^1([0, 1])</math>的函數，<math>\mu</math>是有可數加性的向量测度。

依照上一節的判別基準(*)可以得到以上的結果。

==向量测度的变差==
給定向量测度<math>\mu:\mathcal{F}\to X,</math>，其变差（variation）<math>|\mu|</math>定義如下

: <math>|\mu|(A)=\sup \sum_{i=1}^n \|\mu(A_i)\|</math>

其中{{link-en|最小上界|supremum}}是針對所有<math>\mathcal{F}</math>中<math>A</math>，所有將<math>A</math>[[集合划分|划分]]到有限不交集的划分

: <math>A=\bigcup_{i=1}^n A_i</math>

此處，<math>\|\cdot\|</math>為<math>X</math>的範數。

<math>\mu</math>的变差是有限可加函數，其值在<math>[0, \infty]</math>之間，會使下式成立

: <math>\|\mu(A)\|\le |\mu|(A)</math>

針對任意在<math>\mathcal{F}</math>內的<math>A</math>。若<math>|\mu|(\Omega)</math>是有限的，則测度<math>\mu</math>有有界变差（bounded variation）。可以證明若<math>\mu</math>為具有有界变差的向量测度，則<math>\mu</math>具有可數加性若且唯若<math>|\mu|</math>具有可數加性。
==李亞普諾夫定理==
在向量测度的理論中，{{link-en|阿列克謝·李亞普諾夫|Alexey Lyapunov|李亞普諾夫}}的定理提到non-atomic 向量测度的值域是[[闭集]]及[[凸集]]<ref name="KluvanekKnowles">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&nbsp;J.,&nbsp;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 }}</ref> <!-- Lyapunov's theorem -->。而且non-atomic 向量测度的值域是高维环面（zonoid，是闭集及凸集，是環帶多面體收斂序列的極限）<ref name="DiestelUhl"/>。李亞普諾夫定理有用在[[数理经济学]]<ref name="Aumann" >{{cite journal|first=Robert&nbsp;J.|last=Aumann|title=Existence of competitive equilibrium in markets with a continuum of traders|url=https://archive.org/details/sim_econometrica_1966-01_34_1/page/1|journal=Econometrica|volume=34|number=1|date=January 1966|pages=1–17|jstor=1909854 | mr = 191623|doi=10.2307/1909854}} This paper builds on two papers by Aumann: <p> {{cite journal|<!-- first=Robert&nbsp;J.|last=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}} <p>
<p> {{cite journal|<!-- 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}}  <p></ref><ref>{{cite news|last=Vind|first=Karl|date=May 1964|title=Edgeworth-allocations in an exchange economy with many traders|url=https://archive.org/details/sim_international-economic-review_1964-05_5_2/page/165|journal=International Economic Review|volume=5|pages=165–77|number=2|ref=harv|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|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>、[[起停式控制]]的[[控制理论]]<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"/>及{{link-en|統計理論|statistical theory}}<ref name="Artstein" >{{cite news|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>。
李亞普諾夫定理已可以用[[沙普利－福克曼引理]]證明<ref>{{cite news|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>，後者可以視為是李亞普諾夫定理的[[离散化]]版本<ref name="Artstein" /><ref name="Starr08" >{{cite book|last=Starr|first=Ross&nbsp;M.|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.,&nbsp;ed.|editor2-last=Blume|publisher=Palgrave Macmillan|year=2008|edition=Second|pages=317–318 (1st&nbsp;ed.)|url=http://www.dictionaryofeconomics.com/article?id=pde2008_S000107|doi=10.1057/9780230226203.1518|ref=harv|access-date=2018-12-16|archive-date=2017-03-16|archive-url=https://web.archive.org/web/20170316205747/http://www.dictionaryofeconomics.com/article?id=pde2008_S000107|dead-url=no}}</ref>
<ref>Page 210: {{cite news|last=Mas-Colell|first=Andreu|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>。

==參考資料==
{{Reflist}}

===書籍===
*{{Cite book
  | last = Cohn
  | first = Donald L.
  | title = Measure theory
  | place = Boston&ndash;Basel&ndash;Stuttgart
  | publisher = Birkhäuser Verlag
  | origyear = 1980
  | year = 1997
  | edition = reprint
  | pages = IX+373
  | url = https://books.google.com/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
  
}}
*{{cite book
 | last1      = Diestel
 | first1     = Joe
 | last2      = Uhl
 | first2     = Jerry&nbsp;J.,&nbsp;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
}}
* 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_008|date=1973|publisher=McGraw-Hill|location=New York|page=[https://archive.org/details/functionalanalys00rudi_008/page/n123 114]}}

==相關條目==
*[[博赫纳积分]]

{{泛函分析}}

[[Category:测度]]
[[Category:泛函分析]]
[[Category:控制理论]]