Closure properties of the class of uniform sweeping-out transformations (Q1203483)

The author answers affirmatively a question of \textit{N. A. Friedman} [Can. J. Math. 35, 339-352 (1983; Zbl 0495.28013)] by showing that the class of uniform sweeping-out transformations is closed under countable Cartesian products. A measure-preserving transformation \(S:(X,\mu)\to(X,\mu)\) on a probability space is uniformly sweeping-out if for any measurable set \(A\) with \(\mu(A)>0\) and any \(\varepsilon>0\), there exists \(N\) such that: Any collection \({\mathcal K}\) of integers will satisfy \[ \mu\bigl(\bigcup_{k\in{\mathcal K}}S^ kA\bigr)>1-\varepsilon \] if \(\#{\mathcal K}\geq N\). See Friedman above for a proof that mixing implies uniform sweeping-out, and also Adams (to appear) for an example which is uniform sweeping-out, but is not mixing (or even partial mixing).