Zero-dimensional Hausdorff measure is counting measure

For a finite $A$ of size $N$, its singleton cover costs $N$ at every scale, including $N=0$. If $N\ge 2$, choose $\delta$ smaller than the minimum distance between distinct points; every cover then has at least $N$ members. For $N=1$ any cover needs one member. Thus ${\mathcal{H}}^0(A)=N$.

If $A$ is infinite, for each positive integer $N$ it contains an $N$-point subset. The preceding lower bound gives ${\mathcal{H}}^0(A)\ge N$ for every $N$, hence infinity. The formula is exactly the counting set function.

For any test set $T$ and subset $E$, if $T$ is finite its partition into $T\cap E$ and $T\setminus E$ splits its cardinality. If $T$ is infinite at least one piece is infinite, and both sides of the splitting identity are infinity. This is the Carathéodory criterion for every $E$.