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向量测度

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向量测度(vector measure)是数学名词,是指针对集合族定义的函数,其值为满足特定性质的向量。向量测度是测度概念的推广,测度是针对集合定义的函数,函数的值只有非负的实数

定义及相关推论

给定集合域 巴拿赫空间 有限加性向量测度(finitely additive vector measure)简称测度,是一个满足以下条件的函数:针对任二个内的不交集,下式均成立:

向量测度称为可数加性(countably additive)若针对任意不交集形成的序列 ,都可以让内的联集满足以下条件

等号右边的级数会收敛到巴拿赫空间范数

可以证明向量测度有可数加性,当且仅当针对任何以上的序列,下式均成立

其中的范数。

Σ-代数中定义的可数加性向量测度,会比有限测度(测度的值为非负数)、有限有号测度英语signed measure(测度的值为实数)及复数测度英语complex measure(测度的值为复数)要广泛。

举例

考虑一个由区间的集合形成的场,以及此区间内所有勒贝格测度形成的族。针对任意集合,定义

其中指示函数。依的定义不同,会得到不同的结果。

  • 若是从Lp空间 的函数,是没有可数加性的向量测度。
  • 若是从Lp空间 的函数,是有可数加性的向量测度。

依照上一节的判别基准(*)可以得到以上的结果。

向量测度的变差

给定向量测度,其变差(variation)定义如下

其中最小上界是针对所有,所有将划分到有限不交集的划分

此处,的范数。

的变差是有限可加函数,其值在之间,会使下式成立

针对任意在内的。若是有限的,则测度有有界变差(bounded variation)。可以证明若为具有有界变差的向量测度,则具有可数加性当且仅当具有可数加性。

李亚普诺夫定理

在向量测度的理论中,李亚普诺夫英语Alexey Lyapunov的定理提到non-atomic 向量测度的值域是闭集凸集[1][2][3] 。而且non-atomic 向量测度的值域是高维环面(zonoid,是闭集及凸集,是环带多面体收敛序列的极限)[2]。李亚普诺夫定理有用在数理经济学[4][5]起停式控制控制理论[1][3][6][7]统计理论英语statistical theory[7]。 李亚普诺夫定理已可以用沙普利-福克曼引理证明[8],后者可以视为是李亚普诺夫定理的离散化版本[7][9] [10]

参考资料

  1. ^ 1.0 1.1 Kluvánek, I., Knowles, G., Vector Measures and Control Systems, North-Holland Mathematics Studies 20, Amsterdam, 1976.
  2. ^ 2.0 2.1 Diestel, Joe; Uhl, Jerry J., Jr. Vector measures. Providence, R.I: American Mathematical Society. 1977. ISBN 0-8218-1515-6. 
  3. ^ 3.0 3.1 Rolewicz, Stefan. Functional analysis and control theory: Linear systems. Mathematics and its Applications (East European Series) 29 Translated from the Polish by Ewa Bednarczuk. Dordrecht; Warsaw: D. Reidel Publishing Co.; PWN—Polish Scientific Publishers. 1987: xvi+524. ISBN 90-277-2186-6. MR 0920371. OCLC 13064804. 
  4. ^ Aumann, Robert J. Existence of competitive equilibrium in markets with a continuum of traders. Econometrica. January 1966, 34 (1): 1–17. JSTOR 1909854. MR 0191623. doi:10.2307/1909854.  This paper builds on two papers by Aumann:

    Markets with a continuum of traders. Econometrica. January–April 1964, 32 (1–2): 39–50. JSTOR 1913732. MR 0172689. doi:10.2307/1913732. 

    Integrals of set-valued functions. Journal of Mathematical Analysis and Applications. August 1965, 12 (1): 1–12. MR 0185073. doi:10.1016/0022-247X(65)90049-1. 

  5. ^ Vind, Karl. Edgeworth-allocations in an exchange economy with many traders. International Economic Review 5 (2). May 1964: 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. The Mathematization of economic theory. The American Economic Review. 81, number 1 (Presidential address delivered at the 103rd meeting of the American Economic Association, 29 December 1990, Washington, DC). March 1991: 1–7. JSTOR 2006785. 

  6. ^ Hermes, Henry; LaSalle, Joseph P. Functional analysis and time optimal control. Mathematics in Science and Engineering 56. New York—London: Academic Press. 1969: viii+136. MR 0420366. 
  7. ^ 7.0 7.1 7.2 Artstein, Zvi. Discrete and continuous bang-bang and facial spaces, or: Look for the extreme points. SIAM Review 22 (2). 1980: 172–185. JSTOR 2029960. MR 0564562. doi:10.1137/1022026. 
  8. ^ Tardella, Fabio. A new proof of the Lyapunov convexity theorem. SIAM Journal on Control and Optimization 28 (2). 1990: 478–481. MR 1040471. doi:10.1137/0328026. 
  9. ^ Starr, Ross M. Shapley–Folkman theorem. Durlauf, Steven N.; Blume, Lawrence E., ed. (编). The New Palgrave Dictionary of Economics Second. Palgrave Macmillan. 2008: 317–318 (1st ed.) [2018-12-16]. doi:10.1057/9780230226203.1518. (原始内容存档于2017-03-16). 
  10. ^ Page 210: Mas-Colell, Andreu. A note on the core equivalence theorem: How many blocking coalitions are there?. Journal of Mathematical Economics 5 (3). 1978: 207–215. MR 0514468. doi:10.1016/0304-4068(78)90010-1. 

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向量测度
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