The Hausdorff measure of non-negative dimension $d$ of some set $S$ is:
$$\mathscr{H}^d(S)=\lim_{\delta\to0^+}\inf\left\{\sum_{i=1}^\infty\left(\operatorname{diam}U_i\right)^d:0\le\operatorname{diam}U_i\lt\delta\right\}$$
Where $U$ is drawn from the family of countable covers of the set $S$, so long as each $U_i$ satisfies the diameter constraint. The diameter of each subcover $U_i$ is the greatest distance between any two points in that cover, according to the metric on the metric space.
I gather that the Hausdorff measure is a fairly natural definition of measure; intuitively, considering the set $S$ as some solid shape in $\mathbb{R}^d$ for the sake of intuition, the Hausdorff measure would tell us the volume of that shape. I am aware of the fact that if there exists some $\alpha$ for which $0\lt\mathscr{H}^\alpha(S)\lt\infty$, then $\mathscr{H}^d(S)=0,\,\forall d\gt\alpha,\,\mathscr{H}^d(S)=\infty,\,\forall d\lt\alpha$.
I have two questions. The first is about the summands in the measure: why do we take the sum of all “diameters” to the power of the dimension? I know its naïve and probably not what’s going on in general, but for the sake of learning, I’ll be going with the volume and shape intuition. The diameter power sum would be saying that if you can cover a shape with lots of cubes, since a cube of length $\operatorname{diam}U_i$ will have volume $(\operatorname{diam}U_i)^d$ in $\mathbb{R}^d$, then as these cube covers are pushed to be limiting-ly small they will approach the volume of that shape. However, a cube of that length is absolutely not guaranteed to be the tightest bounding box for some subcover $U_i$ - which begs the question to me, why do we sum with the cube volume formula? With $(\operatorname{diam}U_i)^d$? Couldn’t we just as easily cover our set-shape, for example a unit square in $\mathbb{R}^2$ considered as the set $[0,1]\times[0,1]$, with triangular patches that cover it perfectly with no overlap in the limit?
Second question: how could this be computed? I can see how for a simple set with self similar properties the Hausdorff dimension can be computed - for some fractals you can see how the space it takes up scales as you scales its length, e.g. the Sierpinski triangle which triples in size if you double its length (but knowing the dimension isn’t the same as knowing the measure!), but in general the problem of enumerating all possible countable open covers and then taking their size, with this seemingly arbitrary diameter to the power $d$ sum, and then finding the infimum of that as the covers’ sizes approach zero, seems like a very intractable problem and I wouldn’t know where to start. When Wikipedia says that the Hausdorff dimension of the coastline of Great Britain has been estimated to be one point something, I ask - how? And if so, how could you find its measure?
Many thanks!