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 ,u be a measure on X, v: X -0 X/G _ Y the projection, , r( u). We show that there is a ,-Lusin-measurable disintegration of ,u with respect to v. 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. Introduction. This paper falls naturally into two parts. The first deals with the following situation: G, a compact metric group, acts freely on a locally compact space X (thus, if gx = x for any x E X and g E G, then g = identity in G). The quotient Y = X/ G is locally compact; let '7: X -+ Y be the canonical projection. We show that each measure ,u on X has a 7r( A) = v-Lusin-measurable disintegration with respect to -r (see ?0 for definitions; see [6] for a detailed discussion of disintegrations and their relationship to liftings). No theorem known to the author yields this result, although it is similar to theorems on the disintegration of a measure on a product space (see [2] and [6]). The second part considers a special case: IL is a T-ergodic measure on a compact Hausdorff space X which is the phase space of a bitransformation group (G, X, T) with G metric. Let Go g E Gjf f(gx) di (x) =f(x) du (x) for allf eC(X and let yo be Haar measure on Go. We show that, if y -* is the disintegration of ?1, then each N, looks like yo in a certain sense. The following result is crucial: If Z is a Hausdorff space and f: X -+ Z a ,u-Lusin-measurable, T-invariant map, then f(x) = const ,u-a.e. Finally, in ?6, we remove the metrizability assumption on G; we assume the existence of a strong lifting on (Y, v) (the only place in the paper where this is done). These results represent a portion of the author's Ph.D. thesis, written at Minnesota. The author wishes to thank his advisor, Professor Robert Ellis, Received by the editors April 26, 1976. AMS (MOS) subject classifications (1970). Primary 28A50, 28A65, 54H20. C American Mathematical Society 1977