Topologies on measured groupoids

Abstract

If an analytic Borel group G has a quasiinvariant measure, it is known that G is actually a locally compact group with the original Borel structure being generated by the topology and the original measure being equivalent to Haar measure. In this paper a variation is given on the known proof which then extends to show that an analytic measured groupoid has a σ-compact, and also a locally compact, inessential reduction which is a topological groupoid. In the σ-compact case, it is proved that every “almost” homomorphism agrees a.e. with a (strict) homomorphism. Also, the topology is used to show that every measured groupoid has a complete countable section ¦7¦ and that every locally compact equivalence relation has a complete transversal ¦3¦. These are further used to show that some results of Feldman et al. ¦7¦ apply in general and that a locally compact groupoid with (continuous) Haar system has sufficiently many non-singular Borel G-sets provided that the orbit measures are atom-free ¦23¦.