A graphing of a countable measure-preserving equivalence relation 
on a standard probability
space 
is a countable family of measure-preserving Borel
bijections 
between measurable subsets whose function
graphs lie in 
.
It generates the relation when finite compositions of
these maps and their inverses connect all related points
outside a null set. The graphing has cost 
. Taking the infimum
over generating graphings defines the cost
of an equivalence relation. Thus different graphings can have different costs
while generating the same relation.
Graphing
See also
Cost of an Equivalence Relation, Equivalence Relation, Fixed Price Problem, Probability SpaceExplore with Wolfram|Alpha
References
Anantharaman, C. and Popa, S. An Introduction to
Factors. https://www.math.ucla.edu/~popa/Books/IIun.pdf.
Cite this as:
Weisstein, Eric W. "Graphing." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Graphing.html