Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

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

Author(s): Nieves Castro; Miguel Reyes
Journal: Proc. Amer. Math. Soc. 125 (1997), 3267-3273.
MSC (1991): Primary 28A78
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.

References:

1.: R. Baker, ``Lebesgue measure" on $\mathbb {R}^{\infty }$ , Proc. Amer. Math. Soc. 113 (1991), 1023-1029. MR 92c:46051
2.: K. J. Falconer, The geometry of fractal sets, Cambridge Univ. Press, 1985. MR 88d:28001
3.: C. A. Rogers, Hausdorff measures, Cambridge Univ. Press, 1970. MR 43:7576

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