Hausdorff measure on non separable spaces

Question

In his book Geometry of Sets and Measures in Euclidean Spaces, Pertti Mattila defines the Hausdorff measures via the Carathéodory's construction (chap.4). My doubt is that the Carathéodory's construction is done on a general metric space, while the definition of Hausdorff measure is given starting from a separable space. I personally don't see where separability comes into play. Is it necessary for some reason or we could just keep reasoning on a general metric space?

Answer 1

In your estimates of $s$-dimensional Hausdorff measure of a set $E$, you have to do this: For any $\epsilon > 0$, choose a countable cover $\{U_k\}$ of $E$ with $\text{diam}\; U_k < \epsilon$ for all $k$. Then your estimate is $$ \sum_k \left(\text{diam}\; U_k\right)^s \tag{$*$} $$ You take the infimum over all covers. Then the limit as $\epsilon \to 0$. This is the Hausdorff measure $\mathcal H^s(E)$.

Now suppose $E$ is non-separable. Then for small enough $\epsilon > 0$ there is no countable cover by sets of diameter ${}\lt \epsilon$. Then you have an infimum over the empty set of the numbers ($*$), and we get $+\infty$. And so limit for $\epsilon \to 0$ is also $+\infty$.

Result. For any non-separable set $E$ and for any $s \in [0,+\infty)$, we have Hausdorff measure $\mathcal H^s(E) = +\infty$. And therefore the Hausdorff dimension of $E$ is $+\infty$.

So, if you like, you can define Hausdorff measures for non-separable sets. But it turns out to be uninteresting.