Homotopy and the Kestelman-Borwein-Ditor Theorem
Printed: Mar 2011
N. H. Bingham,
A. J. Ostaszewski,
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.
measure, category, measure-category duality, differentiable homotopy
26A03 - Foundations: limits and generalizations, elementary topology of the line