On the existence of continuous invariant means with respect to amenable semigroups

Some necessary and/or sufficient conditions are investigated, in order to ensure that the action of an amenable semigroup G of homomorphisms on an abstract set X allows invariant means to exist on X. Among our results, there are pointwise ergodic theorems, and extensions of previous results of Tulipani's: in particular, if G is countable, we show that the existence of non-concentrated invariant means implies the existence of continuous ones; moreover we give an example, to show that it is in general not true, if G has higher cardinality.