Baire Measures on Homogeneous Compact Hyper Groups

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Abstract

An infinite compact group is necessarilyuncountable, by the Baire category theorem. A compact "hyper group, inwhich the product of two points is a probability measure, is much like acompact group, having an everywhere supported invariant measure, an orthogonalsystem of characters which span the continuous functions in the uniformtopology, and a multiplicative semigroup of positive-definite functions. It isremarkable that a compact hyper group can be count ably infinite. In this paperhyper groups, which include the algebra of measures on the p-adic integerswhich are invariant under the action of the units (for p = 2, 3, 5, * * * ) ispresented and investigate the question of whether the spectrum or some subsetof it has a hyper group structure.