I was reading Hochman's proof of Frostman's lemma in his online lecture notes here and got hung up. I'm not sure if I'm missing a part of the proof or I'm misunderstanding the theorem itself. The theorem (Theorem 4.11, equivalent to Theorem 4.14) says that if $E \subseteq [0, 1)^{n}$ is compact and $\mathcal{H}^{\alpha}(E) > 0$, then there exists an $\alpha$-regular Borel measure on $E$ that is supported on $E$. To save you the trouble of going through the document, I'll recount the skeleton of the proof up to my question.
We're going to suppose that $F \subseteq \left( \{ 0, 1 \}^{n} \right)^{\mathbb{N}}$ is the representation of $E$ as a family of infinite tree paths on $\left( \{ 0, 1 \}^{n} \right)^{\mathbb{N}}$, related by the map $\pi : \left( d_{k}^{(1)}, \ldots, d_{k}^{(n)} \right)_{k \in \mathbb{N}} \mapsto \left( \sum_{k = 1}^{\infty} d_{k}^{(1)} 2^{-k}, \ldots, d_{k}^{(n)} 2^{-k} \right)$, i.e. $\pi(F) = E$. Let $\mathcal{C}_{\ell}$ denote the $\sigma$-algebra generated by the $\ell$-level cylinders on the sequence space. We will refer to a measure $\mu$ defined on $\mathcal{C}_{\ell}$ as $\ell$-admissible if $$\begin{cases} \mu([d_{1}, \ldots, d_{k} ]) \leq 2^{- \alpha k} & [d_{1}, \ldots, d_{k}] \cap F \neq \emptyset \\ \mu([d_{1}, \ldots, d_{k} ]) = 0 & \textrm{otherwise} \end{cases}$$ for $k = 0, 1, \ldots, \ell$, where the cylinder of the empty string is the entire space $\left( \{ 0, 1 \} \right)^{\mathbb{N}}$. The set of $\ell$-admissible measures can be viewed as vectors in $[0, 1]^{\mathcal{C}_{\ell}}$, and moreover form a complete (thus compact) subset of it so we can use the continuous map $\mu \mapsto \mu \left( \{0, 1\}^{n} \right)^{\mathbb{N}}$ and pick a maximal element according to it (in the sense of maximizes the total measure of the space). Let $\mu_{\ell}$ be a maximal $\ell$-admissible measure. We then view these measures as vectors in $[0, 1]^{\cup_{\ell \geq 1} \mathcal{C}_{\ell}}$, a sequentially compact space, so $(\mu_{\ell})_{\ell \in \mathbb{N}}$ has a convergent subsequence which converges to a finitely additive measure $\mu$ that is $\ell$-admissible for all $\ell$, which we extend to a Borel measure by appealing to Carathedory.
My trouble is with the language of "support". Clearly if a point is not on $F$, then I can find a neighborhood of it that has null measure, so $\operatorname{supp}(\mu) \subseteq F$. But I don't see how it follows that every point of $F$ is in $\operatorname{supp}(\mu)$, that is that every neighborhood of every point of $F$ has nonzero measure. Hochman shows that $\mu(F) > 0$, but it seems as if I could've chosen my $\mu_{\ell}$ such that for a sequence $(d_{k})_{k \in \mathbb{N}} \in F$, I could set $\mu( [d_{1}, \ldots, d_{\ell}] ) = 0$.
I've come to the conclusion that there are two possible solutions to this: (1) I have misunderstood the phrase "$\mu$ supported on $F$" as meaning $\operatorname{supp}(\mu) = F$, when in fact it was intended to mean $\operatorname{supp}(\mu) \subseteq F$; or (2) I missed a detail in this proof on why in fact every neighborhood of every point of $E$ has positive measure.
Could someone please help me with this? Is it a terminological misunderstanding (1) or is it a technical issue (2)? Thanks.