Abstract
Using a new lemma indicated by the title, and a recent measure preserving version of Lusin's Theorem, we prove the following theorem: Any isomorphism-invariant measure theoretic property which is “typical” for automorphisms of a Lebesgue space is also “typical” for Lebesgue measure preserving homeomorphisms of the unit cubeI n ,n≧2. We also prove a partial converse of this theorem. Taken together, these results clarify the relationship between pairs of theorems proved by several authors, which established the “typicality” of specific properties (such as ergodicity and weak mixing) separately in the measurable and continuous cases.
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