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6 - Measure Preserving Lusin Theorem

Published online by Cambridge University Press:  24 August 2009

Steve Alpern
Affiliation:
London School of Economics and Political Science
V. S. Prasad
Affiliation:
University of Massachusetts, Lowell
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Summary

Introduction

A central idea of real variable theory, ‘Littlewood's Second Principle’, is that every measurable function is nearly continuous. Two forms of this principle are contained in the following well known result, the stronger second part of which is known as ‘Lusin's Theorem’.

Theorem 6.1Let g : R → R be a measurable real valued function with |g(x) – x| < ∈ on the interval [a,b]. Then for any δ > 0 there is a continuous function h : R → R with |h(x) – x| < ∈ on [a,b] satisfying

  1. λ {x : | g(x)− h(x)| ≥ δ} < δ, and even

  2. λ {x : g(x)≠ h(x)} < δ.

In this chapter we will prove an analogous result which relates measurable and continuous ergodic theory. That is, we show that a volume preserving bimeasurable bijection of the cube In is nearly a volume preserving homeomorphism. The notion of ‘nearly’ is made precise in the following result obtained by Alpern [8].

Theorem 6.2 (Measure Preserving Lusin Theorem)Let g be a bimeasurable volume preserving bijection (i.e., automorphism) of the cube In, n ≥ 2, with ∥g∥ ≡ ess sup |g(x)−x| < ∈. Then given any δ > 0, there is a volume preserving homeomorphism h of In, with ∥h∥ < ∈ and equal to the identity on the boundary of In, satisfying

  1. λ {x : |g(x)−h(x)| ≥ δ} < δ

  2. λ {x : g(x) ≠ h(x)} < δ.

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Publisher: Cambridge University Press
Print publication year: 2001

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  • Measure Preserving Lusin Theorem
  • Steve Alpern, London School of Economics and Political Science, V. S. Prasad, University of Massachusetts, Lowell
  • Book: Typical Dynamics of Volume Preserving Homeomorphisms
  • Online publication: 24 August 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511543180.008
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  • Measure Preserving Lusin Theorem
  • Steve Alpern, London School of Economics and Political Science, V. S. Prasad, University of Massachusetts, Lowell
  • Book: Typical Dynamics of Volume Preserving Homeomorphisms
  • Online publication: 24 August 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511543180.008
Available formats
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To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Measure Preserving Lusin Theorem
  • Steve Alpern, London School of Economics and Political Science, V. S. Prasad, University of Massachusetts, Lowell
  • Book: Typical Dynamics of Volume Preserving Homeomorphisms
  • Online publication: 24 August 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511543180.008
Available formats
×