  This article is concerned with measure equivalence and uniform measure
equivalence of locally compact, second countable groups. We show that two
unimodular, locally compact, second countable groups are measure equivalent if
and only if they admit free, ergodic, probability measure preserving actions
whose cross section equivalence relations are stably orbit equivalent. Using
this we prove that in the presence of amenability any two such groups are
measure equivalent and that both amenability and property (T) are preserved
under measure equivalence, extending results of Connes-Feldman-Weiss and
Furman. Furthermore, we introduce a notion of uniform measure equivalence for
unimodular, locally compact, second countable groups, and prove that under the
additional assumption of amenability this notion coincides with coarse
equivalence, generalizing results of Shalom and Sauer. Throughout the article
we rigorously treat measure theoretic issues arising in the setting of
non-discrete groups.
