# 向量测度

## 定義及相關推論

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

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

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

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

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

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

## 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數，其值在${\displaystyle [0,\infty ]}$ 之間，會使下式成立

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

## 參考資料

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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)

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