$S(0,M+1)$ together with the union of all balls from $Z^{''\lambda_i}$ centered within $S(0,M+1)^c$. Then if $\{Z^{'\lambda_2}[S(0,M+1)]=\emptyset\}$ occurs and $E_1^{f,M}$ occurs for $Z^{''}$ then $E_2$ occurs for $Z$. So since $Z^{'}$ and $Z^{''}$ are independent we get $$\P[E_2]\ge\P\left[Z^{'\lambda_2}[S(0,M+1)]=\emptyset\right]\P\left[E_1^{f,M}\right]>0$$ which is a contradiction, so $\P[E_1^f]=0$.

Finally we show that $\P[E_1^{\infty}]=0$. However the event $E_1^{\infty}(x)$ is very similar to the event $B^{\infty}(x)$ in the first proof of Theorem \[uniqmon1\], and is shown to have probability $0$ in the same way. In the same way it then follows that $\P[E_1^{\infty}]=0$. [$\Box$ ]{}

Connectivity
============

In this section we show how $\lambda_u$ can be characterized by the connectivity between big balls. This result will be used when we study the model at $\lambda_u$ on a product space in the next section. Let $$\lambda_{BB}:=\inf\{\lambda:\,\lim_{R\rightarrow
  \infty}\inf_{x,y}\P[S(x,R)\leftrightarrow S(y,R)\mbox{ in }C^{\lambda}]=1\}.$$ Note that obviously $\lambda_{BB}\ge \lambda_c$. We will show the following:

\[ballconnect\] For the Poisson Boolean model on a homogeneous space with $\lambda_u<\infty$ we have $\lambda_u=\lambda_{BB}$.

The discrete counterpart of this result is Theorem 3.2 of [@schonmann], and the proof is similar. The proof is also similar to the second proof of Theorem \[uniqmon1\] above. First we show that $\lambda_u\le \lambda_{BB}$.

Suppose that $\lambda_{BB}<\lambda_1<\lambda_2$. We will show that at 