[proofplan] The finite upper $s$-density provides, on uniform level sets $E_m^k$, the bound $\mu(B(x,r)) \leq m \alpha(s) r^s$ for small $r$. For any $t > s$, dividing by $\alpha(t) r^t$ and using $r^{s-t} \to 0$ shows $\Theta^{*t}(\mu, x) = 0$ on each $E_m^k$. The contrapositive of the [Density Implies Dimension Lower Bound](/theorems/3052) then gives $\dim_{\mathcal{H}}(E_m^k) < t$ for every $t > s$, hence $\dim_{\mathcal{H}}(E_m^k) \leq s$. The [Countable Stability of Hausdorff Dimension](/theorems/3066) extends this to $\operatorname{supp}(\mu)$. [/proofplan] [step:Decompose into level sets with uniform density bounds] For $m, k \in \mathbb{N}$, define \begin{align*} E_m^k := \left\{x \in \operatorname{supp}(\mu) \cap \overline{B}(0,k) : \mu(B(x,r)) \leq m \cdot \alpha(s) \cdot r^s \;\text{ for all }\; r \in \left(0, \frac{1}{k}\right)\right\}. \end{align*} Since $\Theta^{*s}(\mu, x) = \limsup_{r \to 0} \mu(B(x,r))/(\alpha(s) r^s) < \infty$ for $\mu$-a.e. $x$, each such $x$ belongs to $E_m^k$ for large enough $m$ and $k$. Therefore $\operatorname{supp}(\mu) = \bigcup_{m,k} E_m^k \cup Z$ where $\mu(Z) = 0$. Since $\mu(Z) = 0$ and every open set meeting $\operatorname{supp}(\mu)$ carries positive $\mu$-mass, the set $\bigcup_{m,k} E_m^k$ is dense in $\operatorname{supp}(\mu)$, so $\operatorname{supp}(\mu) = \overline{\bigcup_{m,k} E_m^k}$. By the [Countable Stability of Hausdorff Dimension](/theorems/3066), it suffices to show $\dim_{\mathcal{H}}(E_m^k) \leq s$ for each $m, k$. [/step] [step:Show $\Theta^{*t}(\mu, x) = 0$ on $E_m^k$ for every $t > s$] Fix $m, k \in \mathbb{N}$ and $t > s$. For every $x \in E_m^k$ and $r \in (0, 1/k)$: \begin{align*} \frac{\mu(B(x,r))}{\alpha(t)\, r^t} \leq \frac{m \cdot \alpha(s) \cdot r^s}{\alpha(t)\, r^t} = \frac{m\, \alpha(s)}{\alpha(t)} \cdot r^{s-t}. \end{align*} Since $s - t < 0$, the right-hand side tends to $0$ as $r \to 0^+$. Therefore \begin{align*} \Theta^{*t}(\mu, x) = \limsup_{r \to 0^+} \frac{\mu(B(x,r))}{\alpha(t)\, r^t} = 0 \quad \text{for every } x \in E_m^k. \end{align*} [/step] [step:Conclude $\dim_{\mathcal{H}}(\operatorname{supp}(\mu)) \leq s$] By the contrapositive of the [Density Implies Dimension Lower Bound](/theorems/3052): if $\dim_{\mathcal{H}}(E_m^k) \geq t$, then there would exist a Borel set $A \subset E_m^k$ with $\mu(A) > 0$ and $\Theta^{*t}(\mu, x) > 0$ for all $x \in A$. But Step 2 shows $\Theta^{*t}(\mu, x) = 0$ for all $x \in E_m^k$, so no such $A$ exists and $\dim_{\mathcal{H}}(E_m^k) < t$. Since this holds for every $t > s$, we conclude $\dim_{\mathcal{H}}(E_m^k) \leq s$. Taking the supremum over all $m, k$: \begin{align*} \dim_{\mathcal{H}}(\operatorname{supp}(\mu)) = \dim_{\mathcal{H}}\!\left(\overline{\textstyle\bigcup_{m,k} E_m^k}\right) = \sup_{m,k} \dim_{\mathcal{H}}(E_m^k) \leq s. \end{align*} [/step]