Given a set X and a semiconvex subadditive measure on it, we establish topological properties of the power set of X, in particular arcwise connectedness, that allow us to investigate the range of signed finitely additive measures (charges), which are strongly non atomic (or equivalently, strongly bounded). In particular we provide the following extension of the Lyapounov Theorem in the signed case: the range of such a finitely additive measure is always convex. On the contrary we provide an example of a charge having non closed range. We also give a partial answer to the question relative to the existence of countably additive restrictions of a charge.
Su alcuni problemi relativi a misure scalari subadditive e applicazioni al caso dell' additività finita
CANDELORO, Domenico;MARTELLOTTI, Anna
1978
Abstract
Given a set X and a semiconvex subadditive measure on it, we establish topological properties of the power set of X, in particular arcwise connectedness, that allow us to investigate the range of signed finitely additive measures (charges), which are strongly non atomic (or equivalently, strongly bounded). In particular we provide the following extension of the Lyapounov Theorem in the signed case: the range of such a finitely additive measure is always convex. On the contrary we provide an example of a charge having non closed range. We also give a partial answer to the question relative to the existence of countably additive restrictions of a charge.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.