u\sP(\eta / u \in O) \geq{}& \mu(O), \quad O \subset C({Q}\oplus L) \text{ open},
 \end{aligned}$$ where $C$ and $O$ are bounded away from $0^K$. Here, $\mu$ is the intensity measure of the PPP $\sum_{i \in {\mathbb N}} \delta_{U_i  F_i(\,\cdot\, -
  T_i)}$ restricted to $C({Q}\oplus L)$. Thus, by adding or removing the boundary, we see that all the limit points of Equation lie in the interval $$\label{eq:liminterval}
   \left[ \frac{\mu(C_B \cap A_L) - \mu(\partial (C_B \cap A_L))}{\mu(A_L) + \mu(\partial A_L)}, \frac{\mu(C_B \cap A_L) + \mu(\partial (C_B \cap A_L))}{\mu(A_L) - \mu(\partial A_L)}\right].$$ We note that $A_L$ is closed and the set $$\begin{aligned}
  A_L^* ={} & \bigg\{ f \in C({Q}\oplus L): \\[-.5em]
  &\quad \, \tau_{Q}(f|_{Q}) \in {Q}^o, \ \max_{t\in {Q}} f(t) > \max\big\{1, f(t)\big\} \ \forall t \in {Q}\oplus L \setminus\{\tau_{Q}(f|_{Q})\} \bigg\}
  \end{aligned}$$ is in the interior of $A_L$ (Lemma \[lem:AL\]). Hence, we can assess $$\begin{aligned}
   \mu(\partial A_L) \leq{} & \quad \ \mu(\{f \in C({Q}\oplus L): \ \max_{t\in {Q}}f(t) = 1\}) \notag\\
                            &      + \mu\bigg( \quad \bigg( \quad \{f \in C({Q}\oplus L): \ \tau_{Q}(f|_{Q}) \in \partial {Q}\} \notag\\
                            & \hspace{1.55cm}  \cup \left\{f \in C({Q}\oplus L): \ \operatorname*{arg\,max}_{t \in {Q}\oplus L} f(t) \text{ is not unique}\right\}\bigg) \notag\\
                            & \qquad \cap \left\{f \in C({Q}\oplus L): \ 