\leq \sum\limits_{v \in \TT } 
   \P_*[S \cap \partial I \neq \emptyset \| v_p = v] = 0$$ $\P$-a.s. thereby completing the proof. $\Cox$

In the remainder of this section, we give the summability criterion that establishes a sufficient condition for absolute continuity in terms of the KL-divergence of the two measures along a ray chosen from $\nu_\TT$.

\[def:splits\] Let $v$ be a vertex of $\TT$ and let $u$ be the parent of $v$. Define $$\begin{aligned}
p(v) & := & \mu_\TT (v)/\mu_\TT (u) \label{p(v)-def}\\
q(v) & := & \nu_\TT (v)/\nu_\TT (u) \label{q(v)-def}\\
X(u) & := & \sum_w q(w) \log [q(w) / p(w)] \label{X-def} \end{aligned}$$

where the sum is over all children $w$ of $u$ and $\nu_\TT(v) = \nu_\TT(\{\gamma : v \in \gamma \})$ and $\mu_\TT(v)$ is defined similarly. The quantity $X$ is known as KL-divergence. The KL-divergence $\KL (\rho,\rho')$ is defined between any two probability measures $\rho$ and $\rho'$ on a finite set $\{ 1 , \ldots , k \}$ by the formula $$\KL (\rho , \rho') := \sum_{i=1}^k \rho'(i) \log \frac{\rho'(i)}{\rho(i)} \, .$$ It is a measure of the difference between the two distributions. It is always non-negative but not symmetric. The following inequality shows that $\KL$ behaves like quadratic 