  Recently, T. Tao gave a finitary proof a convergence theorem for multiple
averages with several commuting transformations and soon later, T. Austin gave
an ergodic proof of the same result. Although we give here one more proof of
the same theorem, this is not the main goal of this paper. Our main concern is
to provide some tools for the case of several commuting transformations,
similar to the tools successfully used in the case of a single transformation,
with the idea that they will be useful in the solution of other problems.

Dynamical Systems