being contained in an infinite open cluster. Thus, conditioned on $\T$ and $v_p$, the law of this subtree is the law of the open cluster containing $v_p$ of Bernoulli($p$) percolation on the subtree below $v_p$ conditioned on this cluster being infinite. Note that since $\TT$ is Galton-Watson, the critical parameter for percolation on the subtree below $v_p$ is $p_c$; in particular, if we call this cluster $C_p$, we have that $\P_*[|C_p| = \infty \| v_p] > 0$ $\nu_\TT$-a.s.

Condition on $\T$ and condition further on $v_p$; note that for $x$ below $v_p$ we have $$\label{eq:union-bd}
\P_*[x \in C_p \| v_p, |C_p| = \infty] 
   \leq \frac{p^{|x| - |v_p|}}{\P_*[|C_p| = \infty \| v_p]}.$$ Thus for any $n$, we have $$\P_*[S \cap \partial I \neq \emptyset \| v_p] 
   \leq \P_*[S_n \leftrightarrow v_p \| v_p,|C_p| = \infty] 
   \leq \frac{|S_n| p^{n-|v_p|}}{\P_*[|C_p| = \infty \| v_p] }$$ where the last inequality is via a union bound over $S_n$. By the definition of $p$, $|S_n|p^n \to 0$; therefore, for each $v_p$, we have $$\P_*[S \cap \partial I \neq \emptyset \| v_p] = 0.$$ Since there are $\P$-a.s. only countably many vertices in $\TT$, the above gives $$\label{eq:mu-eq-zero}
\nu_\TT (S) = \P_*[S \cap \partial I \neq \emptyset ] 
   