The problem of the existence of a countably additive restriction of a finitely additive measure, preserving some properties of the original set function had already been investigated in the scalar and two-dimensional valued cases. In this paper we move to the more general n-dimensional case, by means of a completely different technique and approach. We prove that any bounded finitely additive (n-dimensional) measure m admits a countably additive restriction, having the same range as m., and in addition, that such a restriction can be chosen to be continuous when m is continuous. The result is proved without any additional hypotheses except |Ω|≥c, which is necessary.
R^n-valued finitely additive measures admitting countably additive restrictions with large range
CANDELORO, Domenico;MARTELLOTTI, Anna
1993
Abstract
The problem of the existence of a countably additive restriction of a finitely additive measure, preserving some properties of the original set function had already been investigated in the scalar and two-dimensional valued cases. In this paper we move to the more general n-dimensional case, by means of a completely different technique and approach. We prove that any bounded finitely additive (n-dimensional) measure m admits a countably additive restriction, having the same range as m., and in addition, that such a restriction can be chosen to be continuous when m is continuous. The result is proved without any additional hypotheses except |Ω|≥c, which is necessary.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
