向量测度

向量测度（vector measure）是数学名詞，是指針對集合族定義的函數，其值為滿足特定性質的向量。向量测度是测度概念的推廣，测度是針對集合定義的函數，函數的值只有非負的實數。

定義及相關推論

給定集合域 ${\displaystyle (\Omega ,{\mathcal {F}})}$及巴拿赫空间 ${\displaystyle X}$，有限加性向量測度（finitely additive vector measure）簡稱測度，是一個滿足以下條件的函數${\displaystyle \mu :{\mathcal {F}}\to X}$：針對任二個${\displaystyle {\mathcal {F}}}$內的不交集${\displaystyle A}$和${\displaystyle B}$，下式均成立：

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

向量测度${\displaystyle \mu }$稱為可數加性（countably additive）若針對任意${\displaystyle {\mathcal {F}}}$內不交集形成的序列 ${\displaystyle (A_{i})_{i=1}^{\infty }}$，都可以讓${\displaystyle {\mathcal {F}}}$內的聯集滿足以下條件

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

等號右邊的级数會收斂到巴拿赫空间${\displaystyle X}$的范数。

可以證明向量測度${\displaystyle \mu }$有可數加性，若且唯若針對任何以上的序列${\displaystyle (A_{i})_{i=1}^{\infty }}$，下式均成立

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

其中${\displaystyle \|\cdot \|}$是${\displaystyle X}$的範數。

在Σ-代数中定義的可數加性向量测度，會比有限测度（测度的值為非負數）、有限有號測度（英语：signed measure）（测度的值為實數）及複數测度（英语：complex measure）（测度的值為複數）要廣泛。

舉例

考慮一個由${\displaystyle [0,1]}$區間的集合形成的場，以及此區間內所有勒贝格测度形成的族${\displaystyle {\mathcal {F}}}$。針對任意集合${\displaystyle A}$，定義

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

其中${\displaystyle \chi }$是${\displaystyle A}$的指示函数。依${\displaystyle \mu }$的定義不同，會得到不同的結果。

• ${\displaystyle \mu }$若是從${\displaystyle {\mathcal {F}}}$到Lp空间 ${\displaystyle L^{\infty }([0,1])}$的函數，${\displaystyle \mu }$是沒有可數加性的向量测度。
• ${\displaystyle \mu }$若是從${\displaystyle {\mathcal {F}}}$到L^p空間 ${\displaystyle L^{1}([0,1])}$的函數，${\displaystyle \mu }$是有可數加性的向量测度。

依照上一節的判別基準(*)可以得到以上的結果。

向量测度的变差

給定向量测度${\displaystyle \mu :{\mathcal {F}}\to X,}$，其变差（variation）${\displaystyle |\mu |}$定義如下

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

其中最小上界是針對所有${\displaystyle {\mathcal {F}}}$中${\displaystyle A}$，所有將${\displaystyle A}$划分到有限不交集的划分

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

此處，${\displaystyle \|\cdot \|}$為${\displaystyle X}$的範數。

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

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

針對任意在${\displaystyle {\mathcal {F}}}$內的${\displaystyle A}$。若${\displaystyle |\mu |(\Omega )}$是有限的，則测度${\displaystyle \mu }$有有界变差（bounded variation）。可以證明若${\displaystyle \mu }$為具有有界变差的向量测度，則${\displaystyle \mu }$具有可數加性若且唯若${\displaystyle |\mu |}$具有可數加性。

李亞普諾夫定理

在向量测度的理論中，李亞普諾夫（英语：Alexey Lyapunov）的定理提到non-atomic 向量测度的值域是闭集及凸集^[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，是環帶多面體收斂序列的極限）^[2]。李亞普諾夫定理有用在数理经济学^[4][5]、起停式控制的控制理论^[1][3][6][7]及統計理論（英语：statistical theory）^[7]。 李亞普諾夫定理已可以用沙普利－福克曼引理證明^[8]，後者可以視為是李亞普諾夫定理的离散化版本^{[7][9] [10]}。

參考資料

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 5 (2). May 1964: 165–77. JSTOR 2525560. |journal=被忽略 (帮助) Vind's article was noted by Debreu (1991，第4頁) 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. 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. |journal=被忽略 (帮助)

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 22 (2). 1980: 172–185. JSTOR 2029960. MR 0564562. doi:10.1137/1022026. |journal=被忽略 (帮助)
8. Tardella, Fabio. A new proof of the Lyapunov convexity theorem 28 (2). 1990: 478–481. MR 1040471. doi:10.1137/0328026. |journal=被忽略 (帮助)
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? 5 (3). 1978: 207–215. MR 0514468. doi:10.1016/0304-4068(78)90010-1. |journal=被忽略 (帮助)

書籍

• Cohn, Donald L. Measure theory reprint. Boston–Basel–Stuttgart: Birkhäuser Verlag. 1997: IX+373 [1980]. ISBN 3-7643-3003-1. Zbl 0436.28001.
• Diestel, Joe; Uhl, Jerry J., Jr. Vector measures. Mathematical Surveys 15. Providence, R.I: American Mathematical Society. 1977: 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., Vector measures, Hazewinkel, Michiel (编), 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4
• Rudin, W. Functional analysis. New York: McGraw-Hill. 1973: 114.

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