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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hausdorff measures and dimension on $\mathbb {R}^\infty$
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by Nieves Castro and Miguel Reyes PDF
Proc. Amer. Math. Soc. 125 (1997), 3267-3273 Request permission

Abstract:

We consider the Hausdorff measures $H^{s}$, $0 \leq s < \infty$, defined on $\mathbb {R} ^{\infty } = \prod _{i=1}^{\infty } \mathbb {R}$ with the topology induced by the metric \[ \rho (x,y) = \sum _{i=1}^{\infty } |x_{i}-y_{i}|/2^{i}(1+|x_{i}-y_{i}|),\] for all $x=(x_{i})_{i=1}^{\infty }, y=(y_{i})_{i=1}^{\infty } \in \mathbb {R} ^{\infty }$. We study its properties, their relation to the “Lebesgue measure" defined on $\mathbb {R} ^{\infty }$ by R. Baker in 1991, and the associated Hausdorff dimension. Finally, we give some examples.
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Additional Information
  • Nieves Castro
  • Affiliation: Departamento de Matemática Aplicada, Facultad de Informática, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid, Spain
  • Email: nieves@fi.upm.es
  • Miguel Reyes
  • Affiliation: Departamento de Matemática Aplicada, Facultad de Informática, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid, Spain
  • Email: mreyes@fi.upm.es
  • Received by editor(s): March 25, 1995
  • Received by editor(s) in revised form: May 20, 1996
  • Communicated by: J. Marshall Ash
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 3267-3273
  • MSC (1991): Primary 28A78
  • DOI: https://doi.org/10.1090/S0002-9939-97-03944-0
  • MathSciNet review: 1403116