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

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

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.