  We have shown recently that, given a metric space $X$, the coarse equivalence
classes of metrics on the two copies of $X$ form an inverse semigroup $M(X)$.
Here we give several descriptions of the set $E(M(X))$ of idempotents of this
inverse semigroup and its Stone dual space $\widehat X$. We also construct
$\sigma$-additive measures on $\widehat X$ from finitely additive probability
measures on $X$ that vanish on bounded subsets.
