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The name of George Mackey is sometimes associated with the view that a classifiable equivalence relation is one for which we can assign reals as complete invariants. Recent work in the theory of Borel equivalence relations has tried to understand more generous notions of classification.Additional Reference:
"The stucture of Borel equivalence relations in Polish spaces", by Alexander Kechris, in Set Theory of the Continuum, MSRI Publications Number 26, Springer-Verlag, New-York, 1992.