\mu_T^n$ \[mu-T-def\] exists and is called the [*limit-uniform measure*]{}.

Turning now to Galton-Watson trees, let $\phi (z) := \sum_{n=1}^\infty p_n z^n$ \[phi-def\] be the ordinary generating function for a supercritical branching process with no death, i.e., $\phi (0) = 0$. We recall, $$\begin{aligned}
\phi'  (1) & = & \E Z =: \mu \\ \label{mu-def}
\phi'' (1) & = & \E [Z (Z-1)] \end{aligned}$$ where $Z$ \[Z-def\] is a random variable with probability generating function $\phi$. Throughout, we assume $\E[Z^2] < \infty$; in particular, this also means that $\phi''(1) < \infty$. Moreover, since our focus is on $\partial T$, the assumption of $\phi(0) = 0$ can be made without loss of generality by considering the reduced tree, as in [@athreya-ney Chapter I.12].

We will work on the canonical probability space $(\Omega , \F , \P)$ \[space-def\] where $\Omega = (\N \times [0,1])^{\VU}$, $\F$ is the product Borel $\sigma$-field, and $\P$ is the probability measure making the coordinate functions $\omega_v = (\deg_v , U_v)$ IID with the law of $(Z,U)$, where $U$ is uniform on $[0,1]$ \[U\_v-def\] and independent of $Z$. The variables $\{ \deg_v \}$—where $\deg_v$ \[deg-def\] is interpreted as the number of children of vertex $v$—will construct the Galton-Watson tree, 