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

Let G be a locally compact group with left Haar measure ϒ. The well-known “Theorem LCG” ([10]) states that there is a strong lifting of Mi(G, ϒ) 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 ν0be a Radon measure on Y = X/G, and let µ be the Haar lift of ν0We will show that, if ρ0is a strong lifting of M∞(Y, ν0), then there is a strong lifting M∞(X,µ) which extends ρ0and commutes with the action of G.