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

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.