  In this paper, we prove that if two `box spaces' of two residually finite
groups are coarsely equivalent, then the two groups are `uniform measured
equivalent' (UME). More generally, we prove that if there is a coarse embedding
of one box space into another box space, then there exists a `uniform measured
equivalent embedding' (UME-embedding) of the first group into the second one.
This is a reinforcement of the easier fact that a coarse equivalence (resp.\ a
coarse embedding) between the box spaces gives rise to a coarse equivalence
(resp.\ a coarse embedding) between the groups.
  We deduce new invariants that distinguish box spaces up to coarse embedding
and coarse equivalence. In particular, we obtain that the expanders coming from
$SL_n(\mathbb{Z})$ can not be coarsely embedded inside the expanders of
$SL_m(\mathbb{Z})$, where $n>m$ and $n,m\geq 3$. Moreover, we obtain a
countable class of residually groups which are mutually coarse-equivalent but
any of their box spaces are not coarse-equivalent.
