Sweeping out under a collection of transformations

Subset Generators for Nonsingular Transformations

“…The key to extending the results in §2 to such collections of transformations is extending Theorem 2.1 since the proof of Theorem 2.2 extends without significant change to establish the following result. The following result, proved in [2], shows that the hypothesis of Theorem 4.1 is satisfied by many collections; in particular all infinite subsets of freely acting groups of transformations.…”

“…for T if for every e > 0 there exists B'm& and a positive integer N such that m{B) > 1 -e and T kn B, n ^ N, are pairwise disjoint. 2 and…”

“…For each positive integer n, let C n = f] B t ; hence (1) implies that i = n m ( C n ) > l -2 -" " 1 (2) and lim m(T k iC n ) = 0.…”

“…Call a partition P of X an y-generator if <% is the smallest complete a-algebra (mod m) containing {T s p: p e P, se S}. In §4 we will observe that there arê -generators whenever a certain "sweep-out" property holds, and in [2] it is shown that this property holds for a wide range of y . In particular, if ^ acts freely and y is infinite then ^-generators exist.…”