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February, 1974 Groups of Transformations without Finite Invariant Measures Have Strong Generators of Size 2
Amy J. Kuntz
Ann. Probab. 2(1): 143-146 (February, 1974). DOI: 10.1214/aop/1176996759

Abstract

A size 2 generator of a measure space $(\mathbf{X}, \mathscr{F}, p)$ under a set of $\mathbf{S}$ of transformation of $X$ is a partition $\{A, A^c\}$ of $X$ such that $\mathscr{F}$ is the smallest $\sigma$-algebra containing $\{s^{-1}A: s\in S\}$ up to sets of $p$-measure zero. Let $S$ be a semigroup of invertible nonsingular measurable transformations on a separable measure space $(X, \mathscr{F}, p)$ with $p(X) = 1$. Suppose that $S$ does not preserve any finite invariant measure absolutely continuous with respect to $p$. Then $\mathscr{F}$ has a size 2 generator $\{A, A^c\}$ and the orbit of $A$ under $S$ is dense in $\mathscr{F}$.

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Amy J. Kuntz. "Groups of Transformations without Finite Invariant Measures Have Strong Generators of Size 2." Ann. Probab. 2 (1) 143 - 146, February, 1974. https://doi.org/10.1214/aop/1176996759

Information

Published: February, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0276.28017
MathSciNet: MR355004
Digital Object Identifier: 10.1214/aop/1176996759

Subjects:
Primary: 28A65
Secondary: 20M20

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 1 • February, 1974
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