L. Di Piazza

The McShane, PU and Henstock integrals of Banach valued functions

Some relationships between the vector valued Henstock and McShane integrals are investigated. An integral for vector valued functions, defined by means of partitions of the unity (the PU-integral) is studied. In particular it is shown that a vector valued function is McShane integrable if and only if it is both Pettis and PU-integrable. Convergence theorems for the Henstock variational and the PU integrals are stated. The families of multipliers for the Henstock and the Henstock variational integrals of vector valued functions are characterized.

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.

Selection theorems, based on generalized variation and oscillation

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.

Relations among Gauge and Pettis integrals for cwk(X)- valued multifunctions

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 Role of the Interstitial Oxygen in the Recovery and Evolution of the Boron Implantation Damage

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.

Ereditarietà delle misure di Caratheodory

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.