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.

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

CANDELORO, Domenico;MARTELLOTTI, Anna
1980-01-01

Abstract

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/104786
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