Existence of Borel transversals in groups

Abstract

If M is a closed subgroup of a locally compact group G, we consider the problem of finding a measurable transversal for the cosets G\M — {gM: qeG}—a. measurable subset T c G which meets each coset just once. To each transversal T corresponds a unique cross-section map τ: G/M —> T c G such that π o τ — id, where π: G —> G/M is the canonical mapping. For many purposes it is important to produce reasonably well behaved cross sections for the cosets G/M, and the generality of results obtained is often limited by one's ability to prove that such cross-sections exist. It is well known that, even if G is a connected Lie group, smooth (continuous) cross-sections need not exist; however Mackey ([3], pp. 101-139) showed, using the theory of standard Borel spaces, that a Borel measurable cross section exists if G is a separable (second countable) locally compact group. In this paper topological methods, independent of the theory of standard Borel spaces, are applied to show that Borel measurable cross-sections exist if G is any locally compact group and M any closed subgroup which is metrizable (first countable). The constructions become very simple if G is separable, and give a direct proof that Borel cross-sections exist in this familiar situation.