A “feldman–moore” Representation Theorem for Countable Borel Equivalence Relations on Quotient Spaces

We consider countable Borel equivalence relations on quotient Borel spaces. We prove a generalization of the representation theorem of Feldman–Moore, but provide some examples showing that other very simple properties of countable equivalence relations on standard Borel spaces may fail in the context of nonsmooth quotients. In this paper we try to analyze some very basic questions about the structure of quotients of Polish spaces and Borel equivalence relations modulo another countable equivalence relation; our considerations are motivated by recent work of B. Miller about countable group actions on quotients (which can be found in [9] and [8]). Given a standard Borel space X (see Section 1 for the relevant definitions) and an equivalence relation E on X , the quotient set X/E is a measurable space with respect to the quotient σ -algebra induced by Bor(X), i.e. the largest one making the projection map πE : X → X/E measurable. Unfortunately, this measurable structure is generally not a standard Borel one, so the well-known tools from classical descriptive set theory cannot be applied. In the case of countable Borel E’s, some of the difficulties which arise when dealing with subsets of X/E may be avoided by simply looking at their liftings in the total space X : in Section 1 we define Borel subsets, functions and relations on quotient Borel spaces keeping this idea in mind, and we show that the resulting category differs from the one of measurable (quotient) spaces and measurable maps. In Section 2 we introduce the class of countable Borel equivalence relations on quotient spaces and isolate some interesting subclasses generalizing the usual definitions from the theory of equivalence relations on standard Borel spaces (finite equivalence relations, smooth ones, orbit equivalences associated to group actions); we arrive then at the main question of the paper, the problem of representing countable Borel equivalence relations as orbit equivalences of Borel actions of countable groups. The well-known Feldman–Moore theorem says that every countable Borel equivalence relation on a standard Borel space admits such a representation, and we present a short proof (by Louveau) which well illustrates the two principal ingredients involved: the possibility of “enumerating” the equivalence relation in a definable way and the existence of a countable generating family for the Borel σ -algebra on the space. The second property is no longer valid for quotient spaces, nevertheless we can dispose of it and prove the representation theorem 3, which is the main result of the paper. Finally, in Section 3 we describe some examples showing that a number of easy properties of Borel equivalence relations on standard spaces cease to be true in the context of quotient spaces, and that the additional enumerability hypothesis in theorem