  We answer a question posed by Vitaly Bergelson, showing that in a totally
ergodic system, the average of a product of functions evaluated along
polynomial times, with polynomials of pairwise differing degrees, converges in
$L^{2}$ to the product of the integrals. Such averages are characterized by
nilsystems and so we reduce the problem to one of uniform distribution of
polynomial sequences on nilmanifolds.
