Packing measure

Remarks

The limit in step 1 exists since $P^s_\delta(A)$ decreases as $\delta$ decreases. The map $\mathcal{P}^s$ is a metric outer measure, so it is a measure on the Borel sets. Some authors use the diameters of the balls instead of $2 r_i$; in $\R^n$ the two agree.

Example: on $\R$ with $s = 1$, a $\delta$-packing of $[a, b]$ by intervals has total length at most $b - a + 2\delta$, and packings with total length close to $b - a$ exist. So $P^1_0([a, b]) = b - a$, and in fact $\mathcal{P}^1$ is Lebesgue measure on $\R$.

Step 2 is needed: $P^s_0$ alone is not countably subadditive. The set $\Q \cap [0, 1]$ has $P^1_0 = 1$, because centres in a dense set can approximate any packing of $[0, 1]$. But it is a countable union of points, each with $P^1_0 = 0$, so $\mathcal{P}^1(\Q \cap [0, 1]) = 0$.

For $A \subseteq \R^n$, $\dim_H A \le \dim_P A$, where $\dim_H$ is defined by Hausdorff measure.