The monotone integral of scalar integrand /f /with respect to a vector-valued countably or finitely additive measure /m/ is defined as some form of integration of the distribution function /t //-->//m(f>t)./ Hence in the Banach-valued case, one can think of Bochner or Pettis monotone integral. The usual investigation seeks to compare these form of integration versus the classical approach by means of simple approximation of /f ./It has been proven that the two proposed approaches (Bochner or Pettis) are not suitable, in that one is too strong, and the second too weak. In this paper we introduce a definition for the monotone integral with respect to a Banach-valued finitely additive measure which makes use of the Fremlin-McShane integrability of the function, which is intermediate between the two above. Indeed it turns out that this is the right approach in order to obtain the equivalence of the two theories.
The Monotone integral
SAMBUCINI, Anna Rita;MARTELLOTTI, Anna
1998
Abstract
The monotone integral of scalar integrand /f /with respect to a vector-valued countably or finitely additive measure /m/ is defined as some form of integration of the distribution function /t //-->//m(f>t)./ Hence in the Banach-valued case, one can think of Bochner or Pettis monotone integral. The usual investigation seeks to compare these form of integration versus the classical approach by means of simple approximation of /f ./It has been proven that the two proposed approaches (Bochner or Pettis) are not suitable, in that one is too strong, and the second too weak. In this paper we introduce a definition for the monotone integral with respect to a Banach-valued finitely additive measure which makes use of the Fremlin-McShane integrability of the function, which is intermediate between the two above. Indeed it turns out that this is the right approach in order to obtain the equivalence of the two theories.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
