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Proceedings of the American Mathematical Society

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The approximation of one-one measurable transformations by measure preserving homeomorphisms


Author: H. E. White
Journal: Proc. Amer. Math. Soc. 44 (1974), 391-394
MSC: Primary 28A65
MathSciNet review: 0367159
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Abstract: This paper contains two results related to the material in [2]. Suppose $ f$ is a one-one transformation of the open unit interval $ {I^n}$ (where $ n \geqq 2$) onto $ {I^n}$. 1. If $ f$ is absolutely measureable and $ \varepsilon > 0$, then there is an absolutely measurable homeomorphism $ {\varphi _\varepsilon }$ of $ {I^n}$ onto $ {I^n}$ such that $ m(\{ x:f(x) \ne {\varphi _\varepsilon }(x)$ or $ {f^{ - 1}}(x) \ne \varphi _\varepsilon ^{ - 1}(x)\} ) < \varepsilon $, where $ m$ denotes $ n$-dimensional Lebesgue measure. 2. Suppose $ \mu $ is either (1) a nonatomic, finite Borel measure on $ {I^n}$ such that $ \mu (G) > 0$ for every nonempty open subset $ G$ of $ {I^n}$, or (2) the completion of a measure of type (1). If $ f$ is $ \mu $-measure preserving and $ \varepsilon > 0$, then there is a $ \mu $-measure preserving homeomorphism $ {\varphi _\varepsilon }$ of $ {I^n}$ onto $ {I^n}$ such that $ \mu (\{ x:f(x) \ne {\varphi _\varepsilon }(x)\} ) < \varepsilon $.


References [Enhancements On Off] (What's this?)

  • [1] L. C. Glaser, Geometrical combinatorial topology. Vol. I, Van Nostrand-Reinhold, New York, 1970.
  • [2] Casper Goffman, One-one measurable transformations, Acta Math. 89 (1953), 261–278. MR 0057308
  • [3] J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2) 42 (1941), 874–920. MR 0005803

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DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0367159-0
Keywords: Absolutely measurable transformation, measure preserving transformation, homeomorphism
Article copyright: © Copyright 1974 American Mathematical Society