The measures used to define Hausdorf dimension versus Haar measure

Question

I don't know if there is a name for the measure, so let me construct it: Given a metric space $X$ we define the outer measures $H_\delta^\alpha$ by $\forall A \subset X$

$$H_\delta^\alpha (A)=\inf\lbrace\sum_j r_j^\alpha \mid \exists x_j \in X \text{ such that } A \subset \bigcup_jB(x_j,r_j) \text{ and } \forall j: r_j\le \delta\rbrace$$

We let $H^\alpha(A)=\lim\limits_{\delta\downarrow 0} H_\delta^\alpha(A)$ which is also a metric outer measure. My questions are these:

1. We know that we can restrict the metric outer measure from the power set of $X$ to the Caratheodory measurable sets, and that this results in a genuine measure. It is too much to hope that the Caratheodory measurable sets would be exactly the Borel measurable sets because this is violated even in the Euclidean case, where they differ by the operation of completion relative to the Lebesgue measure. But is it true that at least the measure resulting this way is defined on all of the Borel sets, and possibly more? I.e. are all Borel sets Caratheodory measurable?

2. In a way, this construction can be viewed as a generalization of Lebesgue measure. Haar measure also attempts to be such a generalization. Is there any sense in which these generalizations are related beyond just Lebesgue measure? (This is open ended, but one specific version of this question is: is it true that all locally compact topological groups with abelian group structure that come with a prescribed compatible metric have the Hausdorf measure and the Haar measure equal up to a constant?)

If you are curious why it is in 2. that I insisted on picking one of many compatible metrics to be fixed, then I should mention that I do not know if that step is necessary, as I expressed in this question: Hausdorff Dimension of a manifold of dimension n?

Answer 1

The measure you described is called the spherical Hausdorff measure. It differs from the usual Hausdorff measure in that only coverings by balls are considered.

Both the Hausdorff measure and its spherical version are metric outer measures and therefore Borel sets are measurable with respect to them.

Both of these measures are invariant under isometries of the space, since they are defined purely in terms of metric. Therefore, on a locally compact topological group with [left-] translation-invariant metric they coincide with the [left-] Haar measure up to a constant multiple.