A Note on Locally Compact Groups
نویسنده
چکیده
In this note we shall prove that every locally compact group can be embedded as a closed subgroup in a unimodular group. If the original group is locally Euclidean, the enlarged group will be also, hence the fifth problem of Hilbert is reduced to the unimodular case. We shall use certain results concerning Haar measure whose proof may be found in A. Weil, U integration dans les groupes topologiques et ses applications, Paris, Hermann, 1938, Chap. II, which we shall refer to as AW. Let ƒ stand for a left Haar integral. There is (AW) a continuous function A defined on G such that fGf(xs~)dx=A(s)fo f{x)dx for all integrable functions ƒ and all s £ G .
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