an unbounded component of $C^{\lambda_1}$. Since any ball in $C^{\lambda_1}$ is also present in $C^{\lambda_2}$, this is equivalent to show that any unbounded component of $C^{\lambda_2}$ intersects an unbounded component of $C^{\lambda_1}$. For any point $x\in M$ let $$D(x):=\inf\{d(x,y)\mbox{ : }y\mbox{ is in an unbounded component
    of }C^{\lambda_1}\}$$ and let $$\tilde{D}(x):=\left\{\begin{array}{ll} \inf_{y\in
    C^{\lambda_2}(x)}D(y), & \mbox{if }x\in C^{\lambda_2} \\ D(x), \mbox{ otherwise}
\end{array} \right.$$ Define the random set $H$ to be the set of all points $x$ satisfying the conditions

-   $C^{\lambda_2}(x)$ is a $\lambda_1$-unstable unbounded component

-   $D(x)\le \tilde{D}(x)+1/2$

and write $B(x)$ for the event that $x\in H$. Suppose that $C^{\lambda_2}$ contains an unbounded component which does not intersect an unbounded component of $C^{\lambda_1}$. Then this unbounded component contains regions of positive volume in $H$, so it suffices to show that $\P[B(x)]=0$. Let $H(x)$ be the connected component of $H$ containing $x$. Let $B^{\infty}(x):=B(x)\cap \{\mu(H(x))=\infty\}$ and $B^f
(x):=B(x)\cap\{\mu(H(x))<\infty\}$. The events $B^f$ and $B^{\infty}$ partition $B$. First we show that $\P[B^f(x)]=0$ using the mass transport principle.

In any unbounded component of $C^{\lambda_2}$ not intersecting an unbounded component of $C^{\lambda_1}$ we put mass of unit density. Then all mass in the unbounded component is transported to the regions in the unbounded component 