Cost of an Equivalence Relation

The cost of a countable measure-preserving equivalence relation on a standard probability space is the infimum of the costs of its generating graphings. A graphing is a countable family of measure-preserving Borel bijections between measurable subsets, with each function graph contained in . It generates the relation when finite compositions of the maps and their inverses connect all related points outside a null set. Its cost is

The infimum removes dependence on a particular choice of generating maps.

For a free probability-measure-preserving action of a free group on generators, the orbit equivalence relation has cost . A countable group has fixed price when this cost is the same for all its essentially free probability-measure-preserving actions. An action is essentially free when each nonidentity element fixes a null set.