Inverting the Furstenberg correspondence

Abstract

Given a sequence of subsets A_n of {0,...,n-1}, the Furstenberg correspondence principle provides a shift-invariant measure on Cantor space that encodes combinatorial information about infinitely many of the A_n's. Here it is shown that this process can be inverted, so that for any such measure there are finite sets whose combinatorial properties approximate it arbitarily well. Moreover, we obtain an explicit upper bound on how large n has to be to obtain a sufficiently good approximation. As a consequence of the inversion theorem, we show that every computable invariant measure on Cantor space has a computable generic point. We also present a generalization of the correspondence principle and its inverse to countable discrete amenable groups.