at each square box $\B_{k}$ we put $N_{k}$ points given by $U_{k,1},...,U_{k,N_{k}}$. This procedure determines a Poisson point process $\calP$ from the collections $\calN$ and $\calU_k$ with $k\geq 1$. Given $\e\in\calD_e$ we know that there exist an unique pair $(U_{k,l},U_{m,n})$, where either $m>k$ or $m=k$ and $n>l$, so that $\e=(U_{k,l},U_{m,n})$. Set $\tau_{e}=\tau_{k,l}^{m,n}$.

For each $k\geq 1$ denote by $(\Omega^{k},\mathcal{F}^{k},\bP^{k})$ the probability induced by the random variable $N_k$ and the collections $\calU_k$, $\calT_k$. The probability space $(\Omega,\calF,\bP)$ is defined to be the product space of $(\Omega^{k},\calF^k,\bP^{k})$ over $k\geq 1$.

An important step in the construction of the Busemann function is the proof of the coalescence behavior of semi-infinite geodesics with the same asymptotic orientation. In this proof, we will use the following modification lemma. Let $\mathnormal{K}$ be the collection of all finite sequences $$\label{E:prescription}
 I=\big((k_{j},l_{j},m_{j},n_{j})\big)_{j=1,...,q}\in(\bN^{4})^{q}$$ where $q\geq 1$, $(k_{j},l_{j},m_{j},n_{j})\ne(k_{i},l_{i},m_{i},n_{i})$ for $j\ne i$, $k_{1}\leq...\leq k_{q}$, and either $ k_{j}< m_{j}$ or $l_{j}<n_{j}$. To each $I\in\mathnormal{K}$ corresponds a random vector $(\tau_{k_{j},l_{j}}^{m_{j},n_{j}})_{j=1,...,q}$. We denote $(\Omega_{I},\calF_{I},\bP_{I})$ the probability space induced by this random vector. Let $$\hat{\Omega}_{I}:=\{\hat{\omega}_{I}\,:\,\exists\,\omega_{I}\in\Omega_{I}\mbox{ with }(\hat{\omega}_{I},\omega_{I})\in\Omega\}$$ and denote by $\hat{\bP}_I$ the probability law $\bP$ restricted to this subset. For each $I\subseteq\mathnormal{K}$, $A\subseteq\Omega$ and $\omega_{1}\in\hat{\Omega}_{I}$ define $$A_{I,\omega_{1}}:=\{\omega_{2}\in\Omega_{I}\,:\,\omega=(\omega_{1},\omega_{2})\in A\}\,.$$ Let $\{R_{I}\,:\,I\in\mathnormal{K}\}$ be a family of 