  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.
