Could someone explain the intuition behund the Hausdorff Measure and Hausdorff Dimension?
The Hausdorff Measure is defined as the following:
Let $(X,d)$ be a metric space. $\forall S \subset X$, let $\operatorname{diam} U$ denote the diameter, that is $$\operatorname{diam} U = \sup \{ \rho(x,y) : x,y \in U \} \,\,\,\,\, \operatorname{diam} \emptyset = 0 $$ Let $S$ be any subset of $X$ and $\delta > 0 $ a real number. Define $$ H^{d}_{\delta} (S) := \inf \{ \sum_{i=1}^{\infty} (\operatorname{diam} U_i )^d : \bigcup_{i=1}^{\infty} U_i \supseteq S, \operatorname{diam} U_i < \delta \}$$
What does $\rho(x,y)$ denote? What is meant by the diameter of a set? I'm just trying to understand the intuition behind this definition.
The Hausdorff Dimension is defined as the following:
Let $X$ be a metric space. If $S \subset X$ and $d \in \mathbb{R}^+$, the Hausdorff content is defined as
$$C^{d}_H (S) := \inf \{ \sum_{i} r_i^d : \, \, \text{there is a cover of $S$ by balls of radii} \, \, r_i > 0 \} $$
Then the Hausdorff Dimension is defined as
$$\operatorname{dim}_{H} (X) := \inf \{d \geq 0 : C^d_H = 0\}$$
Could someone explain the intuition behind these definitions?