Vector measure

<h2 id="definitions-and-first-consequences">Definitions and first consequences</h2>
Given a <a href="/facts/Field_of_sets/4E3vQl4f">field of sets</a> 
 
 
 
 (
 Ω
 ,
 
 
 F
 
 
 )
 
 
 {\displaystyle (\Omega ,{\mathcal {F}})}
 
 and a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 a finitely additive vector measure (or measure, for short) is a function 
 
 
 
 μ
 :
 
 
 F
 
 
 →
 X
 
 
 {\displaystyle \mu :{\mathcal {F}}\to X}
 
 such that for any two <a href="/facts/Disjoint_set/nLr8XRVd">disjoint sets</a> 
 
 
 
 A
 
 
 {\displaystyle A}
 
 and 
 
 
 
 B
 
 
 {\displaystyle B}
 
 in 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 one has

μ
        (
        A
        ∪
        B
        )
        =
        μ
        (
        A
        )
        +
        μ
        (
        B
        )
        .
      
    
    {\displaystyle \mu (A\cup B)=\mu (A)+\mu (B).}

A vector measure 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is called countably additive if for any <a href="/facts/Sequence/gV2S4uqp">sequence</a> 
 
 
 
 (
 
 A
 
 i
 
 
 
 )
 
 i
 =
 1
 
 
 ∞
 
 
 
 
 {\displaystyle (A_{i})_{i=1}^{\infty }}
 
 of disjoint sets in 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 such that their union is in 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 it holds that

μ
        
          
            (
            
              
                ⋃
                
                  i
                  =
                  1
                
                
                  ∞
                
              
              
                A
                
                  i
                
              
            
            )
          
        
        =
        
          ∑
          
            i
            =
            1
          
          
            ∞
          
        
        μ
        (
        
          A
          
            i
          
        
        )
      
    
    {\displaystyle \mu {\left(\bigcup _{i=1}^{\infty }A_{i}\right)}=\sum _{i=1}^{\infty }\mu (A_{i})}

with the <a href="/facts/Series_(mathematics)/yDnlsgIe">series</a> on the right-hand side convergent in the <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a> of the Banach space 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

It can be proved that an additive vector measure 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is countably additive if and only if for any sequence 
 
 
 
 (
 
 A
 
 i
 
 
 
 )
 
 i
 =
 1
 
 
 ∞
 
 
 
 
 {\displaystyle (A_{i})_{i=1}^{\infty }}
 
 as above one has

<table><tbody><tr><td> lim n → ∞ ‖ μ ( ⋃ i = n ∞ A i ) ‖ = 0 , {\displaystyle \lim _{n\to \infty }\left\|\mu {\left(\bigcup _{i=n}^{\infty }A_{i}\right)}\right\|=0,} </td> <td></td> <td>*</td></tr></tbody></table>
where 
 
 
 
 ‖
 ⋅
 ‖
 
 
 {\displaystyle \|\cdot \|}
 
 is the norm on 
 
 
 
 X
 .
 
 
 {\displaystyle X.}

Countably additive vector measures defined on <a href="/facts/Sigma-algebra/8LJINLNc">sigma-algebras</a> are more general than finite <a href="/facts/Measure_(mathematics)/yCq7zlI4">measures</a>, finite <a href="/facts/Signed_measure/vukVf84A">signed measures</a>, and <a href="/facts/Complex_measure/zLY1CMtz">complex measures</a>, which are <a href="/facts/Countably_additive_function/yCq7zlI4">countably additive functions</a> taking values respectively on the real interval 
 
 
 
 [
 0
 ,
 ∞
 )
 ,
 
 
 {\displaystyle [0,\infty ),}
 
 the set of <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, and the set of <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>.

<h2 id="examples">Examples</h2>
Consider the field of sets made up of the interval 
 
 
 
 [
 0
 ,
 1
 ]
 
 
 {\displaystyle [0,1]}
 
 together with the family 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 of all <a href="/facts/Lebesgue_measurable_set/8us9cGoW">Lebesgue measurable sets</a> contained in this interval. For any such set 
 
 
 
 A
 ,
 
 
 {\displaystyle A,}
 
 define

μ
        (
        A
        )
        =
        
          χ
          
            A
          
        
      
    
    {\displaystyle \mu (A)=\chi _{A}}

where 
 
 
 
 χ
 
 
 {\displaystyle \chi }
 
 is the <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> of 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
 Depending on where 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is declared to take values, two different outcomes are observed.

<ul><li>
 
 
 
 μ
 ,
 
 
 {\displaystyle \mu ,}
 
 viewed as a function from 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 to the <a href="/facts/Lp_space/gbad0Rv1">
 
 
 
 
 L
 
 p
 
 
 
 
 {\displaystyle L^{p}}
 
-space</a> 
 
 
 
 
 L
 
 ∞
 
 
 (
 [
 0
 ,
 1
 ]
 )
 ,
 
 
 {\displaystyle L^{\infty }([0,1]),}
 
 is a vector measure which is not countably-additive.</li>
<li>
 
 
 
 μ
 ,
 
 
 {\displaystyle \mu ,}
 
 viewed as a function from 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 to the 
 
 
 
 
 L
 
 p
 
 
 
 
 {\displaystyle L^{p}}
 
-space 
 
 
 
 
 L
 
 1
 
 
 (
 [
 0
 ,
 1
 ]
 )
 ,
 
 
 {\displaystyle L^{1}([0,1]),}
 
 is a countably-additive vector measure.</li></ul>
Both of these statements follow quite easily from the criterion (*) stated above.

<h2 id="the-variation-of-a-vector-measure">The variation of a vector measure</h2>
Given a vector measure 
 
 
 
 μ
 :
 
 
 F
 
 
 →
 X
 ,
 
 
 {\displaystyle \mu :{\mathcal {F}}\to X,}
 
 the variation 
 
 
 
 
 |
 
 μ
 
 |
 
 
 
 {\displaystyle |\mu |}
 
 of 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is defined as

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

where the <a href="/facts/Supremum/oXCKVopz">supremum</a> is taken over all the <a href="/facts/Partition_of_a_set/VeP9g8CA">partitions</a>

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

of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 into a finite number of disjoint sets, for all 
  
    
      
        A
      
    
    {\displaystyle A}
  
 in 
  
    
      
        
          
            F
          
        
        .
      
    
    {\displaystyle {\mathcal {F}}.}
  
 Here, 
  
    
      
        ‖
        ⋅
        ‖
      
    
    {\displaystyle \|\cdot \|}
  
 is the norm on 
  
    
      
        X
        .
      
    
    {\displaystyle X.}

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

‖
        μ
        (
        A
        )
        ‖
        ≤
        
          |
        
        μ
        
          |
        
        (
        A
        )
      
    
    {\displaystyle \|\mu (A)\|\leq |\mu |(A)}

for any 
 
 
 
 A
 
 
 {\displaystyle A}
 
 in 
 
 
 
 
 
 F
 
 
 .
 
 
 {\displaystyle {\mathcal {F}}.}
 
 If 
 
 
 
 
 |
 
 μ
 
 |
 
 (
 Ω
 )
 
 
 {\displaystyle |\mu |(\Omega )}
 
 is finite, the measure 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is said to be of bounded variation. One can prove that if 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is a vector measure of bounded variation, then 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is countably additive if and only if 
 
 
 
 
 |
 
 μ
 
 |
 
 
 
 {\displaystyle |\mu |}
 
 is countably additive.

<h2 id="lyapunovs-theorem">Lyapunov's theorem</h2>
In the theory of vector measures, <a href="/facts/Alexey_Lyapunov/KWXSjDhz">Lyapunov</a>'s theorem states that the range of a (<a href="/facts/Atom_(measure_theory)/ojP3UF7Q">non-atomic</a>) finite-dimensional vector measure is <a href="/facts/Closed_set/upYnRTUj">closed</a> and <a href="/facts/Convex_set/vdAuJRJl">convex</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> In fact, the range of a non-atomic vector measure is a <a href="/facts/Zonoid/pQXUI3mv">zonoid</a> (the closed and convex set that is the limit of a convergent sequence of <a href="/facts/Zonotope/w16kH9H3">zonotopes</a>).<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> It is used in <a href="/facts/Mathematical_economics/Dvy7j519">economics</a>,<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> in (<a href="/facts/Bang%25E2%2580%2593bang_control/YpKToXYI">"bang–bang"</a>) <a href="/facts/Control_theory/aIuERpn6">control theory</a>,<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a><a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a><a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> and in <a href="/facts/Statistical_theory/7s5ye1mp">statistical theory</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>
Lyapunov's theorem has been proved by using the <a href="/facts/Shapley%25E2%2580%2593Folkman_lemma/xnko78I9">Shapley–Folkman lemma</a>,<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> which has been viewed as a <a href="/facts/Discretization/JvjSzwyk">discrete</a> <a href="/facts/Discrete_mathematics/XdRUnzUL">analogue</a> of Lyapunov's theorem.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a><a class="footnote-ref" id="fnref:15" href="#fn:15">15</a><a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bochner_measurable_function/HxXFeAgt">Bochner measurable function</a></li>
<li><a href="/facts/Bochner_integral/RpzcTu9n">Bochner integral</a> – Concept in mathematics</li>
<li><a href="/facts/Bochner_space/tSuPabsJ">Bochner space</a> – Type of topological space</li>
<li><a href="/facts/Complex_measure/zLY1CMtz">Complex measure</a> – Measure with complex values</li>
<li><a href="/facts/Signed_measure/vukVf84A">Signed measure</a> – Generalized notion of measure in mathematics</li>
<li><a href="/facts/Vector-valued_functions/qxu8aapt">Vector-valued functions</a> – Function valued in a vector space; typically a real or complex onePages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Weakly_measurable_function/cAFxsUvL">Weakly measurable function</a></li></ul>

<h2 id="bibliography">Bibliography</h2>
<ul><li>Cohn, Donald L. (1997) [1980]. <a href="https://books.google.com/books?id=vRxV2FwJvoAC&q=Measure+theory+Cohn">Measure theory</a> (reprint ed.). Boston–Basel–Stuttgart: <a href="/facts/Birkh%25C3%25A4user_Verlag/brvgnrnJ">Birkhäuser Verlag</a>. pp. IX+373. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-7643-3003-1. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0436.28001">0436.28001</a>.</li>
<li>Diestel, Joe; Uhl, Jerry J. Jr. (1977). Vector measures. Mathematical Surveys. Vol. 15. Providence, R.I: American Mathematical Society. pp. xiii+322. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-1515-6.</li>
<li><a href="/facts/Igor_Kluv%25C3%25A1nek/ac69oJT7">Kluvánek, I.</a>, Knowles, G, Vector Measures and Control Systems, North-Holland Mathematics Studies 20, Amsterdam, 1976.</li>
<li>van Dulst, D. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Vector_measures">"Vector measures"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li>Rudin, W (1973). <a href="https://archive.org/details/functionalanalys00rudi_320">Functional analysis</a>. New York: McGraw-Hill. p. <a href="https://archive.org/details/functionalanalys00rudi_320/page/n123">114</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780070542259.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Kluvánek, I., Knowles, G., Vector Measures and Control Systems, North-Holland Mathematics Studies 20, Amsterdam, 1976. <a href="/wiki/Igor_Kluv%C3%A1nek" target="_blank">/wiki/Igor_Kluv%C3%A1nek</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Diestel, Joe; Uhl, Jerry J. Jr. (1977). Vector measures. Providence, R.I: American Mathematical Society. ISBN 0-8218-1515-6. <a href="0-8218-1515-6" target="_blank">0-8218-1515-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Rolewicz, Stefan (1987). Functional analysis and control theory: Linear systems. Mathematics and its Applications (East European Series). Vol. 29 (Translated from the Polish by Ewa Bednarczuk ed.). Dordrecht; Warsaw: D. Reidel Publishing Co.; PWN—Polish Scientific Publishers. pp. xvi+524. ISBN 90-277-2186-6. MR 0920371. OCLC 13064804. <a href="90-277-2186-6" target="_blank">90-277-2186-6</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Diestel, Joe; Uhl, Jerry J. Jr. (1977). Vector measures. Providence, R.I: American Mathematical Society. ISBN 0-8218-1515-6. <a href="0-8218-1515-6" target="_blank">0-8218-1515-6</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Roberts, John (July 1986). "Large economies". In David M. Kreps; John Roberts; Robert B. Wilson (eds.). Contributions to the New Palgrave (PDF). Research paper. Vol. 892. Palo Alto, CA: Graduate School of Business, Stanford University. pp. 30–35. (Draft of articles for the first edition of New Palgrave Dictionary of Economics). Retrieved 7 February 2011. <a href="/wiki/Donald_John_Roberts" target="_blank">/wiki/Donald_John_Roberts</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Aumann, Robert J. (January 1966). "Existence of competitive equilibrium in markets with a continuum of traders". Econometrica. 34 (1): 1–17. doi:10.2307/1909854. JSTOR 1909854. MR 0191623. S2CID 155044347. This paper builds on two papers by Aumann: Aumann, Robert J. (January–April 1964). "Markets with a continuum of traders". Econometrica. 32 (1–2): 39–50. doi:10.2307/1913732. JSTOR 1913732. MR 0172689.
 Aumann, Robert J. (August 1965). "Integrals of set-valued functions". Journal of Mathematical Analysis and Applications. 12 (1): 1–12. doi:10.1016/0022-247X(65)90049-1. MR 0185073. <a href="/wiki/Robert_Aumann" target="_blank">/wiki/Robert_Aumann</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Vind, Karl (May 1964). "Edgeworth-allocations in an exchange economy with many traders". International Economic Review. Vol. 5, no. 2. pp. 165–77. JSTOR 2525560. Vind's article was noted by Debreu (1991, p. 4) with this comment:

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 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]

Debreu, Gérard (March 1991). "The Mathematization of economic theory". The American Economic Review. Vol. 81, number 1, no. Presidential address delivered at the 103rd meeting of the American Economic Association, 29 December 1990, Washington, DC. pp. 1–7. JSTOR 2006785.
 <a href="/wiki/JSTOR_(identifier)" target="_blank">/wiki/JSTOR_(identifier)</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Kluvánek, I., Knowles, G., Vector Measures and Control Systems, North-Holland Mathematics Studies 20, Amsterdam, 1976. <a href="/wiki/Igor_Kluv%C3%A1nek" target="_blank">/wiki/Igor_Kluv%C3%A1nek</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Rolewicz, Stefan (1987). Functional analysis and control theory: Linear systems. Mathematics and its Applications (East European Series). Vol. 29 (Translated from the Polish by Ewa Bednarczuk ed.). Dordrecht; Warsaw: D. Reidel Publishing Co.; PWN—Polish Scientific Publishers. pp. xvi+524. ISBN 90-277-2186-6. MR 0920371. OCLC 13064804. <a href="90-277-2186-6" target="_blank">90-277-2186-6</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Hermes, Henry; LaSalle, Joseph P. (1969). Functional analysis and time optimal control. Mathematics in Science and Engineering. Vol. 56. New York—London: Academic Press. pp. viii+136. MR 0420366. <a href="/wiki/MR_(identifier)" target="_blank">/wiki/MR_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Tardella, Fabio (1990). "A new proof of the Lyapunov convexity theorem". SIAM Journal on Control and Optimization. 28 (2): 478–481. doi:10.1137/0328026. MR 1040471. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Artstein, Zvi (1980). "Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points". SIAM Review. 22 (2): 172–185. doi:10.1137/1022026. JSTOR 2029960. MR 0564562. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Starr, Ross M. (2008). "Shapley–Folkman theorem". In Durlauf, Steven N.; Blume, Lawrence E. (eds.). The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. pp. 317–318. doi:10.1057/9780230226203.1518. ISBN 978-0-333-78676-5. <a href="978-0-333-78676-5" target="_blank">978-0-333-78676-5</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Page 210: Mas-Colell, Andreu (1978). "A note on the core equivalence theorem: How many blocking coalitions are there?". Journal of Mathematical Economics. 5 (3): 207–215. doi:10.1016/0304-4068(78)90010-1. MR 0514468. <a href="/wiki/Andreu_Mas-Colell" target="_blank">/wiki/Andreu_Mas-Colell</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
</ol>

Vector measure open-in-new

Vector measure