\eta(t), 
   \ \max_{t\in {Q}}\eta(t) \geq a(n)\Big\} \hfill {}={} \hfill \sP\big\{F(\cdot) / \lambda \in B\big\},
 \end{aligned}$$ where $\oplus$ denotes morphological dilation.

The same result holds true if we replace $\limsup_{n \to \infty}$ by $\liminf_{n \to \infty}$.

First, we consider a fixed compact set $L\subset{\mathbb R}^d$ large enough such that $K \cup \{{\bf 0}\} \subset L$ and define $$\begin{aligned}
  A_L = \left\{ f \in C({Q}\oplus L): \ \max_{t\in {Q}}f(t) \geq 1, \ \max_{t\in {Q}}f(t) = \max_{t \in {Q}\oplus L} f(t)\right\}
 \end{aligned}$$ and $$\begin{aligned}
  C_B = \left\{f \in C({Q}\oplus L):
\ f\big(\tau_{Q}(f|_{Q}) + \,\cdot\,\big) \big/ f(\tau_{Q}(f|_{Q})) \in B\right\}
 \end{aligned}$$ for any Borel set $B \subset C(K)$. Note that $C_B$ is invariant w.r.t. multiplication by any positive constant. Thus, we get $$\begin{aligned}
  & \sP\Big\{\eta\big(\tau_{Q}(\eta|_{Q}) + \cdot\big) 
   \big/ \eta(\tau_{Q}(\eta|_{Q})) \in B \ \Big|\  \max_{t\in {Q}}\eta(t) = \max_{t \in {Q}\oplus L} \eta(t)  \geq a(n)\Big\} 
   \nonumber \\
  & ={} \sP\big\{\eta / a(n) \in C_B \,\big|\, \eta / a(n) \in A_L\big\} 
  \nonumber\\
  & ={} \frac{a(n) \sP\big\{\eta/a(n) \in C_B,\,\eta/a(n) 
    \in A_L \big\}}{a(n) \sP\big\{\eta/a(n) \in A_L \big\}}.
  \label{eq:expand-cont}
 \end{aligned}$$

By [@deh2006 Cor. 9.3.2] and [@res2008 Prop. 3.12] we have $$\begin{aligned}
  \limsup_{u \to \infty} u\sP(\eta / u \in C) \leq{}& \mu(C), \quad C \subset C({Q}\oplus L) \text{ closed},\\
  \liminf_{u \to \infty} 