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Homotopy and the Kestelman-Borwein-Ditor Theorem

  Published:2010-08-10
 Printed: Mar 2011
  • N. H. Bingham,
    Mathematics Department, Imperial College London, South Kensington, London SW7 2AZ, U.K.
  • A. J. Ostaszewski,
    Mathematics Department, London School of Economics, Houghton Street, London WC2A 2AE, U.K.
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Abstract

The Kestelman--Borwein--Ditor Theorem, on embedding a null sequence by translation in (measure/category) ``large'' sets has two generalizations. Miller replaces the translated sequence by a ``sequence homotopic to the identity''. The authors, in a previous paper, replace points by functions: a uniform functional null sequence replaces the null sequence, and translation receives a functional form. We give a unified approach to results of this kind. In particular, we show that (i) Miller's homotopy version follows from the functional version, and (ii) the pointwise instance of the functional version follows from Miller's homotopy version.
Keywords: measure, category, measure-category duality, differentiable homotopy measure, category, measure-category duality, differentiable homotopy
MSC Classifications: 26A03 show english descriptions Foundations: limits and generalizations, elementary topology of the line 26A03 - Foundations: limits and generalizations, elementary topology of the line
 

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