[proofplan]
We prove the two-sided density bound by establishing the upper bound $\Theta^{*s}(\mathcal{H}^s \lfloor E, x) \leq 1$ and the lower bound $\Theta_*^s(\mathcal{H}^s \lfloor E, x) \geq 2^{-s}$ separately, each holding $\mathcal{H}^s$-almost everywhere on $E$. The upper bound follows from the [Upper Density Bound](/theorems/3050) applied to the measure $\mathcal{H}^s \lfloor E$. The lower bound follows from the [Lower Density Bound](/theorems/3051). Together they yield $2^{-s} \leq \Theta_*^s \leq \Theta^{*s} \leq 1$ at $\mathcal{H}^s$-a.e. $x \in E$.
[/proofplan]
[step:Recall the density definitions]
Let $\mu = \mathcal{H}^s \lfloor E$ denote the restriction of $\mathcal{H}^s$ to $E$, defined by $\mu(A) = \mathcal{H}^s(A \cap E)$ for every Borel set $A \subset \mathbb{R}^n$. The upper and lower $s$-densities of $\mu$ at a point $x \in \mathbb{R}^n$ are defined by
\begin{align*}
\Theta^{*s}(\mu, x) &= \limsup_{r \to 0^+} \frac{\mu(\overline{B}(x, r))}{\alpha(s) \, r^s}, \\
\Theta_*^s(\mu, x) &= \liminf_{r \to 0^+} \frac{\mu(\overline{B}(x, r))}{\alpha(s) \, r^s},
\end{align*}
where $\alpha(s) = \pi^{s/2} / \Gamma(s/2 + 1)$ is the normalizing constant (the $\mathcal{L}^s$-measure of the unit ball in $\mathbb{R}^{\lfloor s \rfloor}$ when $s$ is an integer). Since $\mathcal{H}^s(E) < \infty$ by hypothesis, $\mu$ is a finite Borel measure on $\mathbb{R}^n$.
[/step]
[step:Establish the upper bound $\Theta^{*s}(\mu, x) \leq 1$ for $\mathcal{H}^s$-a.e. $x \in E$]
By the [Upper Density Bound](/theorems/3050): if $E \subset \mathbb{R}^n$ is a Borel set with $\mathcal{H}^s(E) < \infty$, then
\begin{align*}
\Theta^{*s}(\mathcal{H}^s \lfloor E, x) \leq 1 \quad \text{for } \mathcal{H}^s\text{-a.e. } x \in E.
\end{align*}
We verify the hypothesis: $E$ is $\mathcal{H}^s$-measurable (given) and $\mathcal{H}^s(E) < \infty$ (given). Since $\mathcal{H}^s$ is a Borel regular outer measure by [$\mathcal{H}^s$ is a Borel Regular Outer Measure](/theorems/3044), every $\mathcal{H}^s$-measurable set with finite $\mathcal{H}^s$-measure is contained in a Borel set of equal measure, so the upper density bound applies. The conclusion gives
\begin{align*}
\Theta^{*s}(\mu, x) \leq 1 \quad \text{for } \mathcal{H}^s\text{-a.e. } x \in E.
\end{align*}
[/step]
[step:Establish the lower bound $\Theta_*^s(\mu, x) \geq 2^{-s}$ for $\mathcal{H}^s$-a.e. $x \in E$]
By the [Lower Density Bound](/theorems/3051): if $E \subset \mathbb{R}^n$ is $\mathcal{H}^s$-measurable with $\mathcal{H}^s(E) < \infty$, then
\begin{align*}
\Theta_*^s(\mathcal{H}^s \lfloor E, x) \geq 2^{-s} \quad \text{for } \mathcal{H}^s\text{-a.e. } x \in E.
\end{align*}
Again the hypotheses are verified: $E$ is $\mathcal{H}^s$-measurable and $\mathcal{H}^s(E) < \infty$.
[/step]
[step:Combine the bounds and conclude]
From the previous two steps, both bounds hold $\mathcal{H}^s$-almost everywhere on $E$. Define
\begin{align*}
F_1 &= \{x \in E : \Theta^{*s}(\mu, x) > 1\}, \\
F_2 &= \{x \in E : \Theta_*^s(\mu, x) < 2^{-s}\}.
\end{align*}
Then $\mathcal{H}^s(F_1) = 0$ and $\mathcal{H}^s(F_2) = 0$. For every $x \in E \setminus (F_1 \cup F_2)$ — which is $\mathcal{H}^s$-almost all of $E$ — the inequalities
\begin{align*}
2^{-s} \leq \Theta_*^s(\mu, x) \leq \Theta^{*s}(\mu, x) \leq 1
\end{align*}
hold simultaneously. The middle inequality $\Theta_*^s \leq \Theta^{*s}$ is automatic from the definitions ($\liminf \leq \limsup$ for any function).
[/step]
