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?
Best Answer
It denotes $d(x,y)$. That is, you have a typo somewhere: $d$ and $\rho$ are the same thing, the metric.
What you wrote: the supremum of pairwise distances between the points in your set. To develop intuition, draw a few shapes on the plane (which is an excellent example of a metric space) and determine their diameters. The diameter of a circle is just that, the diameter (hence the name). The diameter of a triangle is the length of its longest side. And so on.
The intuition is that if the object is $d$-dimensional, then $r^d$ roughly represents the volume of its piece of size $r$. Summing over all pieces, we should get something that is no less than the volume of the object. That is, the sums should not be arbitrarily close to zero. And if they are, then the value of $d$ we picked is too high, and the actual dimension of the object is lower than that. So we make $d$ smaller (i.e., take infimum over $d$).
The preceding paragraph is a lie, but this is what you get when you ask for intuition.