A Frostman-type result for measures satisfying uniform lower density conditions Let $\mu$ be a finite, compactly supported, non-zero measure on $\mathbb{R}^d$ for an integer $d$. Let $B(x,r)$ denote the ball of radius $r>0$ centered at $x \in \mathbb{R}^d$. For $\delta \in [0,d]$, we define two conditions: $\mu$ satisfies (C1)-$\delta$ if
$$ \mu(B(x,r)) \leq r^\delta $$
for all $x \in \mathbb{R}^d$ and $r >0$; and $\mu$ satisfies (C2)-$\delta$ if
$$ \mu(B(x,r)) \geq r^\delta$$
for all $r \in (0,1]$ and $x \in \text{supp}(\mu)$, the closed support of $\mu$.
Frostman's Lemma characterizes the sets which support measures satisfying (C1)-$\delta$.
Frostman's Lemma. A Borel set $A \subset \mathbb{R}^d$ supports a non-zero measure satisfying (C1)-$\delta$ if and only if $A$ has positive $\delta$-dimensional Hausdorff measure.
I am interested in a complimentary result concerning measures which satisfy (C2)-$\delta$. Observe that any non-empty set supports a point measure, which satisfies (C2)-$0$ and hence satisfies (C2)-$\delta$ for any $\delta > 0$. So any non-empty set trivially supports a measure satisfying (C2)-$\delta$. The question is rather when a given set is equal to the closed support of such a measure. In this case we only need consider closed sets. We will also restrict to $A$ being compact.
It can be shown using a covering argument that if $A$ is equal to the support of a (C2)-$\delta$ measure, then its Hausdorff dimension is at most $\delta$. This gives a necessary condition. The problem is then as follows.
Problem. Give a sufficient (or necessary and sufficient) condition (e.g. in terms of Hausdorff measure) on a compact set $A \subset \mathbb{R}^d$ which implies the existence of a finite measure $\mu$ satisfying (C2)-$\delta$ such that the closed support of $\mu$ equals $A$.
 A: The equivalence to packing mentioned by Fedja is due to Tricot [1].  See also [2] Sec 5 and [3] sec. 3 for variations and extensions.  The early proofs and most textbooks used dyadic cubes in Euclidean space. But Howroyd  [4], [5] gave the argument in compact metric spaces and this is  included in Mattila's textbook [6]. The packing measure version is in the thesis [7], see also [8].
[1]  C. TRICOT. TWO definitions of fractional dimension. Math. Proc. Camb. Philos. Soc. 91 (1982), 57-74.
[2] S. J. TAYLOR and C. TRICOT. Packing measure, and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288 (1985), 679-699.
[3] Cutler, Colleen D. "Strong and weak duality principles for fractal dimension in Euclidean space." In Mathematical Proceedings of the Cambridge Philosophical Society, vol. 118, no. 3, pp. 393-410. Cambridge University Press, 1995.
[4] Howroyd, John D. "On dimension and on the existence of sets of finite positive Hausdorff measure." Proceedings of the London Mathematical Society 3, no. 3 (1995): 581-604.
[5] Howroyd, John David. "On the theory of Hausdorff measures in metric spaces." PhD diss., UCL (University College London), 1995.
[6] Mattila, Pertti. Geometry of sets and measures in Euclidean spaces: fractals and rectifiability. No. 44. Cambridge university press, 1999.
[7] Joyce, Helen Janeith. "Packing measures, packing dimensions, and the existence of sets of positive finite measure." PhD diss., UCL (University College London), 1995.
[8] Joyce, Helen, and David Preiss. "On the existence of subsets of finite positive packing measure." Mathematika 42, no. 1 (1995): 15-24.
