Luisa Di Piazza

Gauge integrals and selections of weakly compact valued multifunctions

Abstract In the paper Henstock, McShane, Birkhoff and variationally multivalued integrals are studied for multifunctions taking values in the hyperspace of convex and weakly compact subsets of a general Banach space X. In particular the existence of selections integrable in the same sense of the corresponding multifunctions has been considered.

A characterization of the weak Radon-Nikodym property by finitely additive interval functions

A characterization of Banach spaces possessing the weak Radon‐Nikod˝m property is given in terms of finitely additive interval functions. Due to that characterization several Banach space valued set functions that are only finitely additive can be represented as integrals.

A Decomposition of Henstock-Kurzweil-Pettis Integrable Multifunctions

We proved in our earlier paper [9] that in case of separable Banach space-valued multifunctions each Henstock-Kurzweil-Pettis integrable multifunction can be represented as a sum of one of its Henstock-Kurzweil-Pettis integrable selectors and a Pettis integrable multifunction. Now, we prove that the same result can be achieved in case of an arbitrary Banach space. Applying the representation theorem we describe the multipliers of the Henstock-Kurzweil-Pettis integrable multifunctions. Then we use this description to obtain a characterization of the Henstock-Kurzweil-Pettis integrability in terms of subadditive operators.

The Fubini and Tonelli Theorems for Product Local Systems

The notion of product local system and of the Kurzweil-Henstock type integral related to a product local system is introduced. The main result is a version of the Fubini and Tonelli theorems for product local systems.