Idempotent probabilities on semigroups

Summary In this paper, idempotent probability measures have been considered on semigroups which are locally compact or metric and satisfy: (*) A^–1B and Ax^–1 are compact whenever A and B are so, for every x in the semigroup. Such semigroups are more general than compact semigroups which do admit of such measures. On such semigroups we can construct such measures by the usual process if there is a compact sub-semigroup. It is shown in this paper that if such a measure exists in such semigroups, then it must be such an extension measure. Some related results concerning the conditions (*) are also discussed here.