Frostman lemma

Remarks

The direction from $\mu$ to $\mathcal{H}^s(A) > 0$ is the mass distribution principle. If $A \subseteq \bigcup_i U_i$, each nonempty $U_i$ lies in a closed ball of radius $\operatorname{diam} U_i$ centred at a point of $U_i$. So $0 < \mu(A) \le \sum_i \mu(U_i) \le \sum_i (\operatorname{diam} U_i)^s$, and the infimum over covers is at least $\mu(A)$. For the other direction, with $A$ compact, one builds $\mu$ on dyadic cubes from the finest scale upwards, cutting the mass wherever the bound fails, and takes a weak limit. The Borel case reduces to the compact case, since a Borel set with $\mathcal{H}^s(A) > 0$ contains a compact set with positive $\mathcal{H}^s$ measure.

As a consequence, $\dim_H A = \sup \set{ s \ge 0 : \text{a measure as above exists} }$ for every Borel set $A$, where $\dim_H$ is the Hausdorff dimension.

Example: for $A = [0, 1] \subseteq \R$ and $s = 1$, Lebesgue measure restricted to $[0, 1]$ works, since an interval of radius $r$ has length $2r$. After scaling by $\frac12$, it satisfies $\mu(B(x, r)) \le r$.

Going deeper

For a student

Lower bounds for Hausdorff dimension are usually proved by constructing a Frostman measure, since upper bounds come from explicit covers. The lemma leads to the energy method: if $A$ carries a nonzero measure with finite $t$-energy, then $\dim_H A \ge t$. It is a standard tool in projection theorems and in the study of Kakeya sets. It also links Hausdorff dimension to capacity in potential theory.