Benedetto Bongiorno

The essential variation of a function and some convergence theorems

AbstractВВОДИтсь ОпРЕДЕлЕНИ Е ВАРИАцИИ ФУНкцИИ, пР И кОтОРОМ ФОРМУлА

On dyadic integrals and some other integrals associated with local systems

V(F,E) = \int_E {|\bar DF(x)} |dx

APPROXIMATION OF BANACH SPACE VALUED NON-ABSOLUTELY INTEGRABLE FUNCTIONS BY STEP FUNCTIONS

спРАВЕДлИВА Дль пРОИ жВОльНОИ ФУНкцИИF И пРОИжВОльНОгО ИжМЕР ИМОгО МНОжЕстВАE НА ОтРЕжкЕ пРьМОИ. В т ЕРМИНАх ЁтОИ ВАРИАцИ И пОлУЧЕНы тЕОРЕМы О пОЧлЕННОМ ДИФФЕРЕНцИРОВАНИИ п ОслЕДОВАтЕльНОстИ Ф УНкцИИ И тЕОРЕМы О пРЕДЕльНОМ пЕРЕхОДЕ пОД жНАкОМ И НтЕгРАлА ДАНжУА-пЕРР ОНА.

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

Abstract An Henstock–Kurzweil type integral with respect to a P -adic basis is considered. It is shown that a P -adic basis possesses the Ward property if and only if the sequence by which it is defined is bounded. As a consequence, some full descriptive characterizations of the P -adic integral in the bounded case are obtained. Moreover, an example of an exact P -adic primitive which is not a VBG function and does not satisfy the Lusin condition (N) is constructed.

The Scientific Context

We consider some variational properties of indefinite Henstock-type integrals defined by local systems of sets. We study in particular the dyadic path integral which is a special example of such integrals. We obtain full descriptive characterization of the dyadic path integral in terms of the correspondent variational measure and compare it with the Henstock integral associated with the dyadic interval basis.

The Formative Years

We study the fuzzy Henstock and the fuzzy McShane integrals for fuzzy-number valued functions. The main purpose of this paper is to establish the following decomposition theorem: a fuzzy-number valued function is fuzzy Henstock integrable if and only if it can be represented as a sum of a fuzzy McShane integrable fuzzy-number valued function and of a fuzzy Henstock integrable fuzzy number valued function generated by a Henstock integrable function.

The Projects of Guccia: First Stage

The approximation of Banach space valued non-absolutely integrable functions by step functions is studied. It is proved that a Henstock integrable function can be approximated by a sequence of step functions in the Alexiewicz norm, while a Henstock-Kurzweil-Pettis and a Denjoy-Khintchine-Pettis integrable function can be only scalarly approximated in the Alexiewicz norm by a sequence of step functions. In case of Henstock-Kurzweil-Pettis and Denjoy-Khintchine-Pettis integrals the full approximation can be done if and only if the range of the integral is norm relatively compact.

The Projects of Guccia: Second Stage

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.