which are in $H$. Let $\nu(A\times B)$ be the expected mass sent from the set $A$ to the set $B$. Then $\nu$ is easily seen to be a positive diagonally invariant measure on $M\times M$. If $\P[B^f(x)]>0$ then if $A$ is some connected set of finite positive volume, $A$ will get an infinite amount of incoming mass with positive probability, that is $\nu(M\times A)=\infty$. On the other hand, $\nu(A\times M)$, the amount of mass going out from $A$, is at most $\mu(A)<\infty$. Thus by the mass transport principle $\P[B^f(x)]=0$.

Next we show $\P[B^{\infty}(x)]=0$ by showing $\P[B^{\infty}(x)|\tilde{D}(x)=r]=0$ for any $r$. Fix $r$. Suppose $\{\tilde{D}(x)=r\}$ happens. Then for $B^{\infty}(x)$ to happen, there must be infinitely many balls in $C^{\lambda_2}(x)$ centered at distance between $r+1$ and $r+1+1/2$ from unbounded components in $C^{\lambda_1}$. However, this is not possible, as is seen by “building” up the process as follows. Condition on $C^{\lambda_1}$ and then on those balls in $C^{\lambda_2}$ that are centered at distance at least $r+1$ from unbounded components in $C^{\lambda_1}$. We have then not conditioned on the balls that are not present in $C^{\lambda_1}$ but in $C^{\lambda_2}$, and centered at a distance between $0$ and $r+1$ from unbounded components of $C^{\lambda_1}$. These balls 