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.
