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Hausdorff measures and dimension on

Authors:	Nieves Castro Miguel Reyes

Institution:	Departamento de Matemática Aplicada, Facultad de Informática, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid, Spain ; Departamento de Matemática Aplicada, Facultad de Informática, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid, Spain

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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