Disintegration of measures on compact transformation groups

Abstract:

Let G be a compact metrizable group which acts freely on a locally compact Hausdorff space X. Let X, $\mu$ be a measure on $X,\pi :X \to X/G \equiv Y$ the projection, $\nu = \pi (\mu )$. We show that there is a $\nu$-Lusin-measurable disintegration of $\mu$ with respect to it. We use this result to prove a structure theorem concerning T-ergodic measures on bitransformation groups (G, X, T) with G metric and X compact. We finish with some remarks concerning the case when G is not metric.