  We initiate a quantitative study of measure equivalence (and orbit
equivalence) between finitely generated groups, which extends the classical
setting of $\mathrm L^p$ measure equivalence. In this paper, our main focus
will be on amenable groups, for which we prove both rigidity and flexibility
results.
  On the rigidity side, we prove a general monotonicity property satisfied by
the isoperimetric profile, which implies in particular its invariance under
$\mathrm L^1$ measure equivalence. This yields explicit "lower bounds" on how
integrable a measure coupling between two amenable groups can be. This result
also has an unexpected application to geometric group theory: the isoperimetric
profile turns out to be monotonous under coarse embedding between amenable
groups. This has various applications, among which the existence of an
uncountable family of $3$-solvable groups which pairwise do not coarsely embed
into one another.
  On the flexibility side, we construct explicit orbit equivalences between
amenable groups with prescribed integrability conditions. Our main tool is a
new notion of F{\o}lner tiling sequences. We show in a number of instances that
the bounds derived from the isoperimetric profile are sharp up to a logarithmic
factor. We also deduce from this study that two important quasi-isometry
invariants are not preserved under $\mathrm L^1$ orbit equivalence: the
asymptotic dimension and finite presentability.
