we may choose $a$ arbitrary small it follows that $\P[E_2(x)]=0$ as desired.

Next we argue that $\P[E_2(x)]=0$ for all $x$ implies $\P[E_2]=0$. Let $D$ be a countable dense subset of $M$. Then $\P\left[\cup_{x\in
D}E_2(x)\right]=0$. But if $E_2$ occurs then $E_2(x)$ occurs for all $x$ in some unbounded component of $C^{\lambda_2}$, in particular for some $x$ in $D$, so it follows that $\P[E_2(x)]=0$ implies $\P[E_2]=0$.

Next we show that $\P[E_1(x)]=0$. Let $E_1^f(x)$ be the event that $E_1(x)$ occurs and all points in the $\lambda_1$-unstable unbounded $C^{\lambda_2}$-component of $x$ which are at distance less than or equal to three from some unbounded $C^{\lambda_1}$-component are contained in the ball $S(0,N)$ for some random finite $N$. Let $E_1^{\infty}(x)$ be the event that $E_1(x)$ occurs but that there is no such finite $N$. Let $E_1^{f}$ and $E_1^{\infty}$ be the events that $E_1^f(x)$ and $E_1^{\infty}(x)$ respectively happen for some $x$.

First we show that $\P[E_1^f]=0$. Let $E_1^{f,M}:=E_1^f\cap\{N\le
M\}$. We will show that $\P[E_1^f]>0$ implies that $\P[E_2]>0$. So suppose $\P[E_1^f]>0$. Then we may pick $M$ so large that $\P[E_1^{f,M}]>0$. Again let $Z^{'}$ and $Z^{''}$ be independent with the same distribution as $C$. Then for $i=1,2$ let $Z^{\lambda_i}$ be the union of all balls from $Z^{'\lambda_i}$ centered within 