“Vector measure”的意思、由来-开放百科全书

In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only.

Definitions and first consequences

Given a field of sets

and a Banach space

, a finitely additive vector measure (or measure, for short) is a function

such that for any two disjoint sets

and

one has

A vector measure

is called countably additive if for any sequence

of disjoint sets in

such that their union is in

it holds that

with the series on the right-hand side convergent in the norm of the Banach space

It can be proved that an additive vector measure

is countably additive if and only if for any sequence

as above one has

Countably additive vector measures defined on sigma-algebras are more general than finite measures, finite signed measures, and complex measures, which are countably additive functions taking values respectively on the real interval

the set of real numbers, and the set of complex numbers.

Examples

Consider the field of sets made up of the interval

together with the family

of all Lebesgue measurable sets contained in this interval. For any such set

, define

where

is the indicator function of

Depending on where

is declared to take values, we get two different outcomes.

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

The variation of a vector measure

for any

is finite, the measure

is said to be of bounded variation. One can prove that if

is a vector measure of bounded variation, then

is countably additive if and only if

is countably additive.

Lyapunov's theorem

In the theory of vector measures, Lyapunov's theorem states that the range of a (non-atomic) vector measure is closed and convex.^[1]^[2]^[3] 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 zonotopes).^[2] It is used in economics,^[4]^[5]^[6] in ("bang–bang") control theory,^[1]^[3]^[7]^[11] and in statistical theory.^[8]

Lyapunov's theorem has been proved by using the Shapley–Folkman lemma,^[9] which has been viewed as a discrete analogue of Lyapunov's theorem.^[8]^[10]

References

1. ^¹Kluvánek, I., Knowles, G., Vector Measures and Control Systems, North-Holland Mathematics Studies 20, Amsterdam, 1976.
2. ^¹{{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}}
3. ^¹{{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}}}}
4. ^{{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}}}}
5. ^{{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|doi=10.2307/1909854}} 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|doi=10.2307/1913732}}

{{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}}

6. ^{{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|ref=harv|jstor=2525560}} Vind's article was noted by {{harvtxt|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 {{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]

{{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}}
7. ^{{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}}
8. ^¹²{{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}}
9. ^{{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}}
10. ^{{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}}
11. ^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}}

Definitions and first consequences

Examples

The variation of a vector measure

Lyapunov's theorem

References

Books

See also