# 哈尔测度

## 预备知识

• 左变换:
$g S = \{g.s\,:\,s \in S\}$
• 右变换:
$S g = \{s.g\,:\,s \in S\}$

$\mu(g S) = \mu(S). \quad$

## 哈尔定理

• 对任意的g和波莱尔子集E，μ是左变换不变的:
$\mu(gE) = \mu(E)$
• 对所有的紧致集K，μ是有限的:
$\mu(K) <$
$\mu(E) = \inf \{\mu(U): E \subseteq U, U \text{ open and Borel}\}$
• 在波莱尔开集E上μ是内部正则(inner regular)的:
$\mu(E) = \sup \{\mu(K): K \subseteq E, K \text{ compact}\}$

## 右哈尔测度

$\mu_{-1}(S) = \mu(S^{-1}) \quad$

$\mu_{-1}(S g) = \mu((S g)^{-1}) = \mu(g^{-1} S^{-1}) = \mu(S^{-1}) = \mu_{-1}(S). \quad$

$\mu(S^{-1})=k\nu(S)\,$

## 哈尔积分(Harr integral)

$\int_G f(sx) \ d\mu(x) = \int_G f(x) \ d\mu(x)$

## 参考文献

1. ^ 1.0 1.1 Haar, A., Der Massbegriff in der Theorie der kontinuierlichen Gruppen, Annals of Mathematics, 2. 1933, 34 (1): 147–169
2. ^ “外部正则”与“内部正则”是参考日文维基上此条目后翻译出的
3. ^ Weil, André, L'intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles, 869, Paris: Hermann. 1940
4. ^ Alfsen, E.M., A simplified constructive proof of existence and uniqueness of Haar measure, Math. Scand.. 1963, 12: 106–116
• Paul Halmos, Measure Theory, D. van Nostrand and Co., 1950.
• Lynn Loomis, An Introduction to Abstract Harmonic Analysis, D. van Nostrand and Co., 1953.
• André Weil, Basic Number Theory, Academic Press, 1971