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Homotopy and the Kestelman–Borwein–Ditor Theorem

Published online by Cambridge University Press:  20 November 2018

N. H. Bingham  and
A. J. Ostaszewski
N. H. Bingham
Affiliation:
Mathematics Department, Imperial College London, South Kensington, London SW7 2AZ, U.K.e-mail: n.bingham@ic.ac.uk
A. J. Ostaszewski
Affiliation:
Mathematics Department, London School of Economics, Houghton Street, London WC2A 2AE, U.K.e-mail: a.j.ostaszewski@lse.ac.uk
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Abstract

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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 fromthe functional version, and (ii) the pointwise instance of the functional version follows from Miller's homotopy version.

Keywords

26A03measurecategorymeasure-category dualitydifferentiable homotopy
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 1 , 01 March 2011 , pp. 12 - 20
Copyright
Copyright © Canadian Mathematical Society 2011

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