According to the converse of
Haar’s theorem, a complete separable metric group which admits a locally finite
nonzero r∗-invariant measure is locally compact. It is proved in this paper that a
complete separable metric semitopological (resp. topological) semigroup which admits
a finite (resp. possibly infinite) nonzero r∗- and l∗-invariant measure is a compact
(resp. locally compact) topological group.
The structure of idempotent probability measures and finite r∗-invariant measures
has also been given in the case of locally compact semitopological semigroups. Also it
is shown in this paper that for a wide class of nonabelian locally compact
semitopological semigroups, admissibility of a twoside invariant measure is equivalent
to embeddability in a group.
