# 哈尔测度

## 预备知识

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

${\displaystyle \mu (gS)=\mu (S).\quad }$

## 哈尔定理

• 对任意的g和波莱尔子集E，μ是左变换不变的:
${\displaystyle \mu (gE)=\mu (E)}$
• 对所有的紧致集K，μ是有限的:
${\displaystyle \mu (K)<\infty }$
${\displaystyle \mu (E)=\inf\{\mu (U):E\subseteq U,U{\text{ open and Borel}}\}}$
${\displaystyle \mu (E)=\sup\{\mu (K):K\subseteq E,K{\text{ compact}}\}}$

## 右哈尔测度

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

${\displaystyle \mu _{-1}(Sg)=\mu ((Sg)^{-1})=\mu (g^{-1}S^{-1})=\mu (S^{-1})=\mu _{-1}(S).\quad }$

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

## 哈尔积分(Haar integral)

${\displaystyle \int _{G}f(sx)\ d\mu (x)=\int _{G}f(x)\ d\mu (x)}$

## 参考文献

1. Haar, A., Der Massbegriff in der Theorie der kontinuierlichen Gruppen, Annals of Mathematics, 2 34 (1), 1933, 34 (1): 147–169, JSTOR 1968346
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 the existence and uniqueness of Haar measure, Math. Scand., 1963, 12: 106–116 [2020-03-25], （原始内容存档于2020-11-26）
• 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