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In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933. Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, and ergodic theory. ==Preliminaries== Let (''G,.'') be a locally compact Hausdorff topological group. The σ-algebra generated by all open sets of ''G'' is called the Borel algebra. An element of the Borel algebra is called a Borel set. If ''g'' is an element of ''G'' and ''S'' is a subset of ''G'', then we define the left and right translates of ''S'' as follows: * Left translate: :: * Right translate: :: Left and right translates map Borel sets into Borel sets. A measure μ on the Borel subsets of ''G'' is called ''left-translation-invariant'' if for all Borel subsets ''S'' of ''G'' and all ''g'' in ''G'' one has : A similar definition is made for right translation invariance. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Haar measure」の詳細全文を読む スポンサード リンク
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