Vol. 25, No. 3, 1968

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Existence of Borel transversals in groups

Jacob Feldman and Frederick Paul Greenleaf

Vol. 25 (1968), No. 3, 455–461

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 : q G}—a measurable subset T G which meets each coset just once. To each transversal T corresponds a unique cross-section map τ : G∕M T G such that π 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.

Mathematical Subject Classification
Primary: 22.20
Received: 7 July 1967
Published: 1 June 1968
Jacob Feldman
Univ of California, Berkeley
Berkeley CA
United States
Frederick Paul Greenleaf