Movatterモバイル変換

Vector measure

From Wikipedia, the free encyclopedia

Inmathematics, avector measure is afunction defined on afamily of sets and takingvector values satisfying certain properties. It is a generalization of the concept of finitemeasure, which takesnonnegative real values only.

Definitions and first consequences

[edit]

Given afield of sets $(\Omega ,{\mathcal {F}})$ and aBanach space $X, {\displaystyle X,}$ afinitely additive vector measure (ormeasure, for short) is a function $\mu :{\mathcal {F}}\to X$ such that for any twodisjoint sets $A {\displaystyle A}$ and $B {\displaystyle B}$ in ${\mathcal {F}}$ one has $\mu (A\cup B)=\mu (A)+\mu (B).$

A vector measure $\mu$ is calledcountably additive if for anysequence $(A_{i})_{i=1}^{\infty }$ of disjoint sets in ${\mathcal {F}}$ such that their union is in ${\mathcal {F}}$ it holds that $\mu {\left(\bigcup _{i=1}^{\infty }A_{i}\right)}=\sum _{i=1}^{\infty }\mu (A_{i})$ with theseries on the right-hand side convergent in thenorm of the Banach space $X . {\displaystyle X.}$

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

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

where $\|\cdot \|$ is the norm on $X . {\displaystyle X.}$

Countably additive vector measures defined onsigma-algebras are more general than finitemeasures, finitesigned measures, andcomplex measures, which arecountably additive functions taking values respectively on the real interval $[0,\infty ),$ the set ofreal numbers, and the set ofcomplex numbers.

Examples

[edit]

Consider the field of sets made up of the interval $[0,1]$ together with the family ${\mathcal {F}}$ of allLebesgue measurable sets contained in this interval. For any such set $A, {\displaystyle A,}$ define $\mu (A)=\chi _{A}$ where $\chi$ is theindicator function of $A . {\displaystyle A.}$ Depending on where $\mu$ is declared to take values, two different outcomes are observed.

$\mu ,$ viewed as a function from ${\mathcal {F}}$ to the $L^{p}$ -space $L^{\infty }([0,1]),$ is a vector measure which is not countably-additive.
$\mu ,$ viewed as a function from ${\mathcal {F}}$ to the $L^{p}$ -space $L^{1}([0,1]),$ is a countably-additive vector measure.

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

The variation of a vector measure

[edit]

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

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

Lyapunov's theorem

[edit]

In the theory of vector measures,Lyapunov's theorem states that the range of a (non-atomic) finite-dimensional vector measure isclosed andconvex.^[1]^[2]^[3] In fact, the range of a non-atomic vector measure is azonoid (the closed and convex set that is the limit of a convergent sequence ofzonotopes).^[2] It is used ineconomics,^[4]^[5]^[6] in ("bang–bang")control theory,^[1]^[3]^[7]^[8] and instatistical theory.^[8]Lyapunov's theorem has been proved by using theShapley–Folkman lemma,^[9] which has been viewed as adiscrete analogue of Lyapunov's theorem.^[8]^[10]^[11]

References

[edit]

^^a ^bKluvánek, I., Knowles, G.,Vector Measures and Control Systems, North-Holland Mathematics Studies 20, Amsterdam, 1976.
^^a ^bDiestel, Joe; Uhl, Jerry J. Jr. (1977).Vector measures. Providence, R.I: American Mathematical Society.ISBN 0-8218-1515-6.
^^a ^bRolewicz, 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.
^Roberts, John (July 1986). "Large economies". InDavid M. Kreps;John Roberts;Robert B. Wilson (eds.).Contributions to theNew 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 ofNew Palgrave Dictionary of Economics). Retrieved7 February 2011.
^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.
^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 byDebreu (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 andif 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, seeVind (1964)."] But explanations of the ... functions of prices ... can be made to rest on theconvexity of sets derived by that averaging process.Convexity in the commodity spaceobtained 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.
^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 ^b ^cArtstein, 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.
^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.
^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.
^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.

Bibliography

[edit]

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.

v t e Analysis intopological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector-valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner /Weakly /Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold

Functional analysis (topics –glossary)

Spaces

Banach Besov Fréchet Hilbert Hölder Nuclear Orlicz Schwartz Sobolev Topological vector
Properties	Barrelled Complete Dual (Algebraic /Topological) Locally convex Reflexive Separable

Theorems

Operators

Algebras

Open problems

Applications

Advanced topics

Category

Measure theory

Basic concepts

Sets

Types ofmeasures

Particular measures

Maps

Main results

Carathéodory's extension theorem
Convergence theorems
Decomposition theorems
Egorov's
Fatou's lemma
Fubini's
- Fubini–Tonelli
Hölder's inequality
Minkowski inequality
Radon–Nikodym
Riesz–Markov–Kakutani representation theorem

Other results

Disintegration theorem Lifting theory Lebesgue's density theorem Lebesgue differentiation theorem Sard's theorem Vitali–Hahn–Saks theorem
ForLebesgue measure	Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality