From Gales to Covers
This post continues to study relationship between definitions of Hausdorff dimension obtained through covers and gales. In the last post, we have shown how to switch from covers to gales, which allowed us to conclude that \[ \inf G(X) \le \dim_H(X) \] So now our aim should be verification of reverse inequality, i.e. $\dim_H(X)\le \inf G(X)$. In order to do this, we go back to definitions. Let us start with basics $\mathcal{C} = \{0,1\}^{\mathbb{N}}$ is Cantor space with some subset $X$. Let $d$ be an $s$-gale which succeeds on $X$, i.e. $X\subseteq S^{\infty}[d]$. This means that $\inf G(X)\le s$. Furthermore, for any given $\epsilon>0$, one could choose $s$ such that $\inf G(X)\ge s-\epsilon$. Given such $s$, we claim that $\mathcal{H}^s(X)=0$, i.e. \[ \inf\{\sum_{i=1}^{\infty}diam(C_{w_i})^s \} = 0 \] where $X\subseteq \bigcup C_{w_i}$. This would give us $\dim_H(X)\le s \le \inf G(X) + \epsilon$. By letting $\epsilon\to 0$, the desired inequality follows. So our main...