events $R_{I}\in\calF_{I}$ such that $\bP_{I}(R_{I})>0$ for all $I$. Then define the map on $\calF$ by $$\Phi_{I}(A):=\{\omega_{1}\in\hat{\Omega}_{I}\,:\,\bP_{I}(A_{I,\omega_{1}})>0\}\times R_{I}.$$ Suppose that $W(\omega)$ is a random element of $\mathnormal{K}$, which may be interpreted as the set of indexes (edges) whose passage time value will be modified. For $A\subseteq\Omega$, let $$\tilde{\Phi}(A):=\cup_{I\in\mathnormal{K}}\big[\{\omega_{1}\in\hat{\Omega}_{I}\,:\,A(I)_{I,\omega_{1}}\ne\emptyset\}\times R_{I}\big],$$ where $A(I):=A\cap [W=I]$.

\[lmodif\] For each $A\in\calF$, $\tilde{\Phi}(A)$ contains $\Phi(A)\in\calF$ defined as the following union $$\Phi(A):=\cup_{I\in\mathnormal{K}}\Phi_{I}\big(A(I)\big).$$ Furthermore, if $\bP(A)>0$ then $\bP\big(\Phi(A)\big)>0$.

If $\bP_{I}\big(A(I)_{I,w_{1}}\big)>0$ then $A(I)_{I,w_{1}}\ne\emptyset$ and so $\Phi(A)\subseteq\tilde{\Phi(A)}$. Since $\mathnormal{K}$ is countable and $A=\cup_{I\in\mathnormal{K}}A(I)$, if $\bP(A)>0$ then there exists $I\in\mathnormal{K}$ such that $\bP\big(A(I)\big)>0$. For this $I$, by Fubini’s theorem $$\label{emodi2} 
 0<\bP\big(A(I)\big)=\int_{\hat{\Omega}_{I}}\bP_{I}\big(A(I)_{I,w_{1}}\big)\hat{\bP}_{I}(dw_{1}).$$ Let $$\hat{A}_{I}:=\{ w_{1}\,:\,\bP_{I}\big(A(I)_{I,w_{1}}\big)>0\}\,.$$ By , $\hat{\bP}_{I}\big(\hat{A}_{I}\big)>0$. According to the definition of $\Phi_{I}$, $$\bP\Big(\Phi_{I}\big(A(I)\big)\Big)=\hat{\bP}_{I}(\hat{A}_{I})\bP_{I}(R_{I})>0\,.$$ Since $\Phi_{I}\big(A(I)\big)\subseteq\Phi(A)$, we conclude that $\bP\big(\Phi(A)\big)>0$.

Some geometrical aspects of Delaunay triangulations {#pre-geom}
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In this part we are going to study some geometrical aspects of Delaunay triangulations. Let $\x,\y\in\bR^2$ and construct a path $\gamma(\x,\y):=(\v_1,...,\v_{k})$ in $\calD$ connecting $\v(\x)$ to $\v(\y)$ as follows: set $\v_1:=\v(\x)$; if $\v_1\neq\v(\y)$ let $\v_{2}$ be the (almost-surely) unique nearest neighbor of $\v_1$ such that the edge of $\C_{\v_1}$ that is perpendicular to the line segment $[\v_1,\v_2]$ cross $[\x,\y]$; given $\v_l$ with $l\geq 1$, if $\v_{l}\neq\v(\y)$ then we set $\v_{l+1}$ to be the (almost-surely) 