  The first $l^p$-cohomology is an algebro-analytical object attached to a
finitely generated discrete group and introduced by M. Gromov. It is well known
that it is invariant under quasi-isometry. In this article, we prove that the
non-vanishing of the first $l^p$-cohomology of a non-amenable group is
invariant under $L^q$-Measured Equiavalence (an equivalence relation introduced
by Gromov), where $q\geq p$. We also discuss many applications of this result.
We prove that for hyperbolic (in the sense of Gromov) Coxeter groups with
boundaries having Combinatorial Loewner Property, conformal dimension (of the
canonical conformal gauge) of the Gromov boundary is invariant under
$L^q$-Measured Equivalence for some large $q$. We prove that the finitely
generated free groups and surface groups are not $L^1$-Measured Equivalent. We
also give a lower bound of the critical exponent for the first $l^p$-cohomology
of any lattice in $SO(n,1)$. Finally, we discuss $L^q$-Measured Equivalence
between non-amenable 3-manifold groups corresponding to Thurston's three
geometries $\mathbb{H}^3$, $\mathbb{H}^2\times\mathbb{R}$ and
$\widetilde{SL_2(\mathbb{R})}$.
