  We study a positive-definite function associated to a measure-preserving
equivalence relation on a standard probability space and use it to measure
quantitatively the proximity of subequivalence relations. This is combined with
a recent co-inducing construction of Epstein to produce new kinds of mixing
actions of an arbitrary infinite discrete group and it is also used to show
that orbit equivalence of free, measure preserving, mixing actions of
non-amenable groups is unclassifiable in a strong sense. Finally, in the case
of property (T) groups we discuss connections with invariant percolation on
Cayley graphs and the calculation of costs.
