Disintegration of Measures on Compact Transformation Groups

To prove 1.1, one first assumes X is compact and G is a Lie group. In this case, X is "measure-theoretically" the product Y x G; this follows from the existence of local cross-sections to the projection n [6]. Let n2 : X ~ Y x G —> G, and define a map £ from L(Y, v) to the space of Radon measures on G as follows: £(ƒ) = TÏ2 [if ° n) ' M] • Apply the Dunford-Pettis Theorem [3] to ? to obtain a map co from Y to M+(G) = the set of positive Radon measures 17 on G such that ||T?|| = 1. The map X is easily obtained from co. One now completes the proof by (i) approximating G by a sequence of Lie groups [6] ; (ii) using the fact that there is a locally countable collection of pairwise disjoint compact subsets of Y the complement of whose union is locally i>-null [1].