\max_{t\in {Q}}f(t) = \max_{t \in {Q}\oplus L} f(t) \geq 1 \right\}\bigg) \notag \\
                     \leq{} & 0 + \int_{\partial {Q}} \int_{\lambda^{-1}}^\infty u^{-2} \sd u \sd t_0 \notag\\
                            & \phantom{0} + \int_{{\mathbb R}^d \setminus ({Q}\oplus L)} \int_{\lambda^{-1}}^\infty u^{-2} \sP\left\{u  \max_{t_0 \in {Q}} F(t_0 -x) \geq 1\right\} \sd u \sd x \label{eq:partAL}.
  \end{aligned}$$ Here, the equality $\mu(\{f \in C({Q}\oplus L): \ \max_{t\in {Q}}f(t) = 1\}) = 0$ holds as $\max_{t \in {Q}} M(t)$ is Fréchet distributed (cf. [@deh2006 Lemma 9.3.4]). Since $\partial {Q}$ is a Lebesgue null set, the second term on the right-hand side of also vanishes. Thus, $$\begin{aligned}
   \mu(\partial A_L) \leq{} & \int_{{\mathbb R}^d \setminus ({Q}\oplus L)} \int_{\lambda^{-1}}^\infty u^{-2} \sP\left\{u  \max_{t_0 \in {Q}} F(t_0 -x) \geq 1\right\} \sd u \sd x =: c(L) \label{eq:partAL2}.
  \end{aligned}$$

Now, let $B \subset C(K)$ a be Borel set such that $\sP(F / \lambda \in \partial B) = 0$. For the set $C_B$, we obtain that the set $$\begin{aligned}
     C_B^* ={} & \bigg\{ f \in C({Q}\oplus L): \ \operatorname*{arg\,max}_{f \in {Q}} f(t) \text{ is unique},\  \frac{f\big(\tau_{Q}(f|_{Q}) + \cdot\big)}{f(\tau_{Q}(f|_{Q}))} \in B^o \bigg\}
  \end{aligned}$$ is in the interior of $C_B$ and that the closure of $C_B$ is a subset of $$\begin{aligned}
   C_B^* \cup{} & \left\{f \in C({Q}\oplus L): 