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Proceedings of the American Mathematical Society

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

 

 

Hausdorff measures and dimension on $\mathbb {R}^{\infty }$


Authors: Nieves Castro and Miguel Reyes
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
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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

\begin{displaymath}\rho (x,y) = \sum _{i=1}^{\infty } |x_{i}-y_{i}|/2^{i}(1+|x_{i}-y_{i}|),\end{displaymath}

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

DOI: https://doi.org/10.1090/S0002-9939-97-03944-0
Received by editor(s): March 25, 1995
Received by editor(s) in revised form: May 20, 1996
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1997 American Mathematical Society