Disintegration of measures on compact transformation groups

Let G be a compact metrizable group which acts freely on a locally compact Hausdorff space X. Let μ be a measure on X, it: X -* X/G s Y the projection, p=π(μ). We show that there is ar-Lusin-measurable disintegration of p with respect to n. 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.