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In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure. This allows one to define integrals of Borel measurable functions on ''G'' so that standard analysis notions such as the Fourier transform and spaces can be generalized. Many of the results of finite group representation theory are proved by averaging over the group. For compact groups, modifications of these proofs yields similar results by averaging with respect to the normalized Haar integral. In the general locally compact setting, such techniques need not hold. The resulting theory is a central part of harmonic analysis. The representation theory for locally compact abelian groups is described by Pontryagin duality. ==Examples and counterexamples== *Any compact group is locally compact. *Any discrete group is locally compact. The theory of locally compact groups therefore encompasses the theory of ordinary groups since any group can be given the discrete topology. *Lie groups, which are locally Euclidean, are all locally compact groups. *A Hausdorff topological vector space is locally compact if and only if it is finite-dimensional. *The additive group of rational numbers Q is not locally compact if given the relative topology as a subset of the real numbers. It is locally compact if given the discrete topology. *The additive group of ''p''-adic numbers Q''p'' is locally compact for any prime number ''p''. 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「Locally compact group」の詳細全文を読む スポンサード リンク
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