The definition of the Hausdorff measure $H_s(S)$ of a subset $S$ of $\mathbb{R}^n$ is $\lim_{\delta\to 0} H_s^\delta(S)$ (where $H_s^\delta(S)$ is the infimum of all $\sum_n diam(U_n)^s$ where $(U_n)$ is an countable open covering of $S$ such that $diam(U_n)<\delta$ for all $n$.
My question is the following :
Is there a reason as to why we restrict the diameter of the $U_n$ by a number $\delta$ and then take the limit when $\delta\to 0$ ?
In other words, couldn't we just define a measure $H'_s(S)$ by saying that $H'_s(S)$ is the infimum of all $\sum_n diam(U_n)^s$ where $(U_n)$ is a countable open covering of $S$ ?
Would that give a measure, and would that give something interesting?