On mixing in infinite measure spaces |
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Authors: | U. Krengel L. Sucheston |
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Affiliation: | (1) Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, 43210 Columbus, Ohio, USA |
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Abstract: | Summary Two concepts of mixing for null-preserving transformations are introduced, both coinciding with (strong) mixing if there is a finite invariant measure. The authors believe to offer the correct answer to the old problem of defining mixing in infinite measure spaces. A sequence of sets is called semiremotely trivial if every subsequence contains a further subsequence with trivial remote -algebra (=tail -field). A transformation T is called mixing if (T–nA) is semiremotely trivial for every set A of finite measure; completely mixing if this is true for every measurable A. Thus defined mixing is exactly the condition needed to generalize certain theorems holding in finite measure case. For invertible non-singular transformations complete mixing implies the existence of a finite equivalent invariant mixing measure. If no such measure exists, complete mixing implies that for any two probability measures 1,2, in total variation norm.Research of this author is supported by the National Science Foundation (U.S.A.) under grant GP 7693. |
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