I have two questions on basic properties of the Hausdorff measure and dimension which I've taken for granted for a while (I'm revisiting Falconer after about a year), but that I've never actually seen shown.
Firstly, I've always seen the Hausdorff dimension defined as: $$\dim_{H}(F) = \inf \{ s \geq 0 : H^{s}(F) = 0 \} = \sup \{ s \geq 0 : H^{s}(F) = \infty \}$$ and been shown the graph of the measure going from $\infty$ to $0$ at a critical point $s$, but I've never actually seen the proof that these two values actually coincide, i.e. that this graph is correct.
Also, I've never actually seen it shown that the Hausdorff measure is countably additive on the Borel $\sigma$-algebra under disjoint union (at least on $\mathbb{R}^{n}$), i.e. that it is indeed a measure.
If anybody could demonstrate these to me, or point me in the right direction, I'd greatly appreciate it.
The fact that Hausdorff measure is a Borel measure follows fairly easily from the fact that it is a metric outer measure, i.e. $${\cal H}^s(A \cup B) = {\cal H}^s(A) + {\cal H}^s(B),$$ whenever $$\inf_{a\in A,b\in B} \text{dist}(a,b)>0.$$ I think the fact that ${\cal H}^s$ is a metric outer measure is fairly easy to prove from the definition. After all, $${\cal H}^s(E) = \lim_{\varepsilon \rightarrow 0^+} {\cal H}^s_{\varepsilon}(E)$$ and the equality should hold for small enough $\varepsilon$. From here, it's not hard to see that ${\cal H}^s$ should be additive on closed sets, from which it follows that it should be additive on Borel sets. The details must certainly be worked out in the classic text Hausdorff Measures by C. A. Rogers.