On the norm convergence of non-conventional ergodic averages

We offer a proof of the following non-conventional ergodic theorem: If T(i): Z(r) curved right arrow (X, Sigma, mu) for i = 1, 2,..., d are commuting probability-preserving Z(r)-actions, (I(N))(N >= 1) is a FOlner sequence of subsets of Z(r), (a(N))(N >= 1) is a base-point sequence in Z(r) and f(1), f(2),..., f(d) is an element of L(infinity)(mu) then the non-conventional ergodic averages 1/vertical bar I(N)vertical bar Sigma n is an element of I(N)+a(N) i=1 Pi d f(i) o T(i)(n) converge to some limit in L(2)(mu) that does not depend on the choice of (a(N))(N >= 1) or (I(N))(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 Tao's 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.