  Tao has recently proved that if $T_1,...,T_l$ are commuting, invertible,
measure-preserving transformations on a dynamical system then for any
$L^\infty$ functions $f_1,...,f_l$, the average
$\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i\leq l}f_i\circ T^n_i$ converges in the
$L^2$ norm. Tao's proof is unusual in that it translates the problem into a
more complicated statement about the combinatorics of finite spaces by using
the Furstenberg correspondence "backwards". In this paper, we give an ergodic
proof of this theorem, essentially a translation of Tao's argument to the
ergodic setting. In order to do this, we develop two new variations on the
usual Furstenberg correspondence, both of which take recurrence-type statements
in one dynamical system and give equivalent statements in a different dynamical
system with desirable properties.
