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# 向量测度

## 定义及相关推论

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

${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }A_{i}\right)=\sum _{i=1}^{\infty }\mu (A_{i})}$

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

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

${\displaystyle \mu (A)=\chi _{A))$

• ${\displaystyle \mu }$若是从${\displaystyle {\mathcal {F))}$Lp空间 ${\displaystyle L^{\infty }([0,1])}$的函数，${\displaystyle \mu }$是没有可数加性的向量测度。
• ${\displaystyle \mu }$若是从${\displaystyle {\mathcal {F))}$Lp空间 ${\displaystyle L^{1}([0,1])}$的函数，${\displaystyle \mu }$是有可数加性的向量测度。

## 向量测度的变差

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

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

${\displaystyle \mu }$的变差是有限可加函数，其值在${\displaystyle [0,\infty ]}$之间，会使下式成立

${\displaystyle \|\mu (A)\|\leq |\mu |(A)}$

## 参考资料

1. Kluvánek, I., Knowles, G., Vector Measures and Control Systems, North-Holland Mathematics Studies 20, Amsterdam, 1976.
2. Diestel, Joe; Uhl, Jerry J., Jr. Vector measures. Providence, R.I: American Mathematical Society. 1977. ISBN 0-8218-1515-6.
3. 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. 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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