  We get three types of results on measure equivalence rigidity; direct product
groups of Ozawa's class $\mathcal{S}$ groups, wreath product groups and
amalgamated free products. We prove measure equivalence factorization results
on direct product groups of Ozawa's class $\mathcal{S}$ groups. As
consequences, Monod--Shalom type orbit equivalence rigidity theorems follow. We
prove that if two wreath product groups $A \wr G$, $B \wr \Gamma$ of
non-amenable exact direct product groups $G$, $\Gamma$ with amenable bases $A$,
$B$ are measure equivalent, then $G$ and $\Gamma$ are measure equivalent. We
get Bass--Serre rigidity results on amalgamated free products of non-amenable
exact direct product groups.
