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.
