## 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.## References

- Richard Baker,
*“Lebesgue measure” on $\textbf {R}^\infty$*, Proc. Amer. Math. Soc.**113**(1991), no. 4, 1023–1029. MR**1062827**, DOI 10.1090/S0002-9939-1991-1062827-X - K. J. Falconer,
*The geometry of fractal sets*, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR**867284** - C. A. Rogers,
*Hausdorff measures*, Cambridge University Press, London-New York, 1970. MR**0281862**

## 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