  We offer a new proof of the Furstenberg-Katznelson multiple recurrence
theorem for several commuting probability-preserving transformations T_1, T_2,
>..., T_d: \bbZ\curvearrowright (X,\S,\mu), and so, via the Furstenberg
correspondence principle introduced in, a new proof of the multi-dimensional
Szemeredi Theorem. We bypass the careful manipulation of certain towers of
factors of a probability-preserving system that underlies the
Furstenberg-Katznelson analysis, instead modifying an approach recently
developed for the study of convergence of nonconventional ergodic averages to
pass to a large extension of our original system in which this analysis greatly
simplifies. The proof is then completed using an adaptation of arguments
developed by Tao for his study of an infinitary analog of the hypergraph
removal lemma. In a sense, this addresses the difficulty, highlighted by Tao,
of establishing a direct connection between his infinitary, probabilistic
approach to the hypergraph removal lemma and the infinitary, ergodic-theoretic
approach to Szemeredi's Theorem set in motion by Furstenberg.
