On the norm convergence of non-conventional ergodic averages

We offer a proof of the following non-conventional ergodic theorem: If Ti:^rℤ^r(X,μ,μ) for i=1,2,⋯,d are commuting probability-preserving ^r-actions, (IN)N1 is a Flner sequence of subsets of ℤ^r, (IN)N≥1 is a base-point sequence in ℤ^r and f1,f2,⋯,fd∈L∞(μ) then the non-conventional ergodic averages 1/|IN| ∑n∈IN+aN ∏i=1 converge to some limit in L²(μ) that does not depend on the choice of (aN)N≥1 or (IN)N≥1. The leading case of this result, with r=1 and the standard sequence of averaging sets, was first proved by Tao, following earlier analyses of various more special cases and related results by Conze and Lesigne, Furstenberg and Weiss, Zhang, Host and Kra, Frantzikinakis and Kra and Ziegler. While Taos proof rests on a conversion to a finitary problem, we invoke only techniques from classical ergodic theory, so giving a new proof of his result.