Continuous finitely additive measures which are extensions of non-atomic measures

We face the problem of the existence of countably additive restrictions of a strongly non atomic scalar finitely additive measure, defined on the whole power set of an abstract set X.The mere existence problem was already solved in the literature; however these earlier results gave restrictions that did not preserve some desirable properties of the assigned set functions: for instance its range or the strong non atomicity. In the paper a deeper set of results is given in two different cases: when X is uncountable, it is proven that a non atomic countably additive restriction to an algebra always exist, preserving the whole range, while the case of countable support X is pathological in that no continuous non null measure can be defined on a sigma algebra of the set of the integers.