real analysis - Question regarding Lebesgue outer measure.

Given $m\geq1$, $0\leq s<\infty$, $0<\delta\leq\infty$ and $A\subseteq\mathbb{R}^{m}$ define:

$$\mathcal{H}_{\delta}^{s}\left(A\right)=\inf\left\{ {\displaystyle \sum_{n=1}^{\infty}d\left(B_{n}\right)^{s}\,|\,\left\{ B_{n}\right\} _{n\in\mathbb{N}}}\:\mbox{is a covering of }A\:\mbox{and }d\left(B_{n}\right)<\delta\,\forall\, n\geq1\right\}$$
For ease of notation this refers to countable coverings but it also includes finite coverings. Now I've shown the following two things:

1. For all $m\geq1$, $0\leq s<\infty$, $0<\delta\leq\infty$ this defines an external measure on $\mathbb{R}^{m}$.


2. For all $0\leq s<\infty$ and $A\subseteq\mathbb{R}^{m}$ the limit $\mathcal{H}^{s}\left(A\right)=\lim\limits _{\delta\downarrow0}\mathcal{H}_{0}^{s}\left(A\right)$ exists.

I'm now trying to show these following two things:

1. For all $0\leq s<\infty\quad$ $\mathcal{H}^{s}$ is an external metric measure on $\mathbb{R}^{m}$.


2. $\mathcal{H}^{0}$ is the counting measure on $\mathbb{R}^{m}$

Here is what I know so far regarding each of these goals:

1. I want to show that for each $A,B\subseteq\mathbb{R}^{m}$ if $dist(A,B)>0$ then $$\mathcal{H}^{s}(A\cup B)=\mathcal{H}^{s}(A)+\mathcal{H}^{s}(B)$$ I know that $dist(A,B)>0$ is equivalent to $\overline{A}\cap\overline{B}=\emptyset$ so I assume I should use that somehow but I'm not sure how.


2. I've shown it works for finite sets but I'm having trouble with infinite sets. Obviously for $s=0$ we get that $\mathcal{H}_{\delta}^{0}\left(A\right)$ is a function only of the number of sets in the covering for which the infima is obtained. What I want to show is that given an infinite subset $A\subseteq\mathbb{R}^{m}$ as $\delta\downarrow0$ the number of sets required in order to cover $A$ by sets of diameter less than $\delta$ goes to infinity.
   At first this struck me as a bit odd since for a compact infinite set we know that for each $\delta>0$ there is a finite $\delta$-net covering $A$ but then I realized that doesn't mean the number of sets in these nets doesn't go to infinity.

Anyway, I'd really appreciate some thick hints or a full proof of these two claims, I've already spent a couple of hours wrecking my head over it alone :)

Answer

For (2) - for any $n$, an infinite set $A$ contains a subset of cardinality $n$. So as $H^0$ is an outer measure, $H^0(A)\geq n$. As this holds for all $n$, we are done.