Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28A65

Retrieve articles in all journals with MSC: 28A65


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0367159-0
PII: S 0002-9939(1974)0367159-0
Keywords: Absolutely measurable transformation, measure preserving transformation, homeomorphism
Article copyright: © Copyright 1974 American Mathematical Society