As stated above, I'd like to prove that the 0-dimensional Hausdorff Measure of a set $F \subset \mathbb{R}^n$ is the cardinality of $F$. In other words, that $\mathscr{H}^0 (F) = |F|$, or the number of points in $F$.
I was wondering if I could allow $|U_i|$ to be a $\delta-$cover of each point in the set, but I have no idea really where to go with this. Direction would be very helpful. I am not seeking a full proof.
Let $F$ be a finite set in $\mathbb R^n$ and let $\varepsilon = \min\{|x-y|:x,y\in F\}$. Then, if $\delta<\varepsilon/2$, we have $\mathscr{H}_{\delta}^0(F)=|F|$. This is because we certainly need exactly one set $U$ in an open $\delta$-cover of $F$ for each point in $F$ and, for each such set, $\text{diam}(F)^0=1$. If $F$ is infinite, I think it's not too hard to show that $\mathscr{H}^0(F)=\infty$.