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Analytic computable structure theory and $L^p$ spaces

Volume 244 / 2019

Joe Clanin, Timothy H. McNicholl, Don M. Stull Fundamenta Mathematicae 244 (2019), 255-285 MSC: Primary 03D45; Secondary 03D78, 46B04. DOI: 10.4064/fm448-5-2018 Published online: 9 November 2018

Abstract

We continue the investigation of analytic spaces from the perspective of computable structure theory. We show that if $p \geq 1$ is a computable real, and if $\Omega $ is a nonzero, nonatomic, and separable measure space, then every computable presentation of $L^p(\Omega )$ is computably linearly isometric to the standard computable presentation of $L^p[0,1]$; in particular, $L^p[0,1]$ is computably categorical. We also show that there is a measure space $\Omega $ that does not have a computable presentation even though $L^p(\Omega )$ does for every computable real $p \geq 1$.

Authors

  • Joe ClaninDepartment of Computer Science
    Iowa State University
    Ames, IA 50011, U.S.A.
    e-mail
  • Timothy H. McNichollDepartment of Mathematics
    Iowa State University
    Ames, IA 50011, U.S.A.
    e-mail
  • Don M. StullDepartment of Computer Science
    Iowa State University
    Ames, IA 50011, U.S.A.
    and
    Laboratoire Lorrain de recherche
    en informatique et ses applications
    Campus scientifique
    BP 239
    54506 Vandœuvre-lès-Nancy Cedex, France
    e-mail
    e-mail

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