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.
Factors.