Vol. 76, No. 1, 1978

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Existence of a strong lifting commuting with a compact group of transformations

Russell Allan Johnson

Vol. 76 (1978), No. 1, 69–81

Let G be a locally compact group with left Haar measure γ. The well-known “Theorem LCG” ([10]) states that there is a strong lifting of M(G,γ) commuting with left translations. We will prove partial generalizations of this theorem in case G is compact. Thus, let (G,X) be a free (left) transformation group with G, X compact such that (I) G is abelian, or (II) G is Lie, or (III) X is a product G × Y . Let ν0 be a Radon measure on Y = X∕G, and let μ be the Haar lift of ν0. We will show that, if ρ0 is a strong lifting of M(Y,ν0), then there is a strong lifting M(X,μ) which extends ρ0 and commutes with the action of G.

Mathematical Subject Classification 2000
Primary: 28A65, 28A65
Secondary: 22D40
Received: 23 September 1977
Published: 1 May 1978
Russell Allan Johnson