The Projects of Guccia: Second Stage

G.B. Guccia had an ambitious goal: developing the Circolo Matematico di Palermo into the international association of mathematicians. The pursuit of this aim was influenced by his two most important mathematical relationships: with Vito Volterra and with Henri Poincare. The time came to obtain a professorship, and to consolidate the Circolo Matematico di Palermo. On both issues, G.B. Guccia succeeded, although through rather turbulent processes. The celebration of the Heidelberg International Congress of 1904 was a fundamental moment for the international expansion of the Circolo Matematico di Palermo. The Rome International Congress of 1908 gave G.B. Guccia the opportunity to realize his a…

The Scientific Context

We review three important processes: the foundation of national mathematical societies, in particular, the London Mathematical Society and the Societe Mathematique de France; the creation of research mathematical journals, focusing on Acta Mathematica and its founder Gosta Mittag-Leffler; and the organization of mathematics at international level, discussing the origins of the International Congresses of Mathematicians.

A variational henstock integral characterization of the radon-nikodým property

A characterization of Banach spaces possessing the Radon-Nikodym property is given in terms of finitely additive interval functions. We prove that a Banach space X has the RNP if and only if each X-valued finitely additive interval function possessing absolutely continuous variational measure is a variational Henstock integral of an X-valued function. Due to that characterization several X-valued set functions that are only finitely additive can be represented as integrals.

The Projects of Guccia: First Stage

The founding of the Circolo Matematico di Palermo and its journal, the Rendiconti del Circolo Matematico di Palermo are discussed. In order to understand the path that led to these two events, we follow G.B. Guccia’s post-doctoral journey in the summer of 1880 through Paris, Reims and London. Despite many initial difficulties, the early success of the society and the journal encouraged G.B. Guccia to lead the society towards internationalization.

The Formative Years

The school and university education of G.B. Guccia are presented. He studied in the new educational system created when the Kingdom of Italy was founded. Sicily had a long story of cultivators of mathematics, which we briefly review. Shortly after G.B. Guccia entered university the most important encounter of his life occurred: he met the geometer Luigi Cremona and moved to Rome. Five years later he presented his thesis under the guidance of Cremona.

On the Minimal Solution of the Problem of Primitives

Abstract We characterize the primitives of the minimal extension of the Lebesgue integral which also integrates the derivatives of differentiable functions (called the C -integral). Then we prove that each BV function is a multiplier for the C -integral and that the product of a derivative and a BV function is a derivative modulo a Lebesgue integrable function having arbitrarily small L 1 -norm.

The Riesz Representation Theorem and Extension of Vector Valued Additive Measures

Radon-Nikodym derivatives of finitely additive interval measures taking values in a Banach space with basis

Let X be a Banach space with a Schauder basis {en}, and let Φ(I)= ∑n en ∫I fn(t)dt be a finitely additive interval measure on the unit interval [0, 1], where the integrals are taken in the sense of Henstock–Kurzweil. Necessary and sufficient conditions are given for Φ to be the indefinite integral of a Henstock–Kurzweil–Pettis (or Henstock, or variational Henstock) integrable function f:[0, 1] → X.

On Variational Measures Related to Some Bases

Abstract We extend, to a certain class of differentiation bases, some results on the variational measure and the δ-variation obtained earlier for the full interval basis. In particular the theorem stating that the variational measure generated by an interval function is σ-finite whenever it is absolutely continuous with respect to the Lebesgue measure is extended to any Busemann–Feller basis.

A Constructive Minimal Integral which Includes Lebesgue Integrable Functions and Derivatives

In this paper we provide a minimal constructive integration process of Riemann type which includes the Lebesgue integral and also integrates the derivatives of differentiable functions. We provide a new solution to the classical problem of recovering a function from its derivative by integration, which, unlike the solution provided by Denjoy, Perron and many others, does not possess the generality which is not needed for this purpose.The descriptive version of the problem was treated by A. M. Bruckner, R. J. Fleissner and J. Foran in [2]. Their approach was based on the trivial observation that for the required minimal integral, a function F is the indefinite integral of f if and only if F'…

A Decomposition Theorem for the Fuzzy Henstock Integral

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 essential variation of a function and some convergence theorems

ВВОДИтсь ОпРЕДЕлЕНИ Е ВАРИАцИИ ФУНкцИИ, пР И кОтОРОМ ФОРМУлА $$V(F,E) = \int_E {|\bar DF(x)} |dx$$ 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–NIKODÝM PROPERTY BY FINITELY ADDITIVE INTERVAL FUNCTIONS

AbstractA characterization of Banach spaces possessing the weak Radon–Nikodým 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.

Differentiation of an additive interval measure with values in a conjugate Banach space

We present a complete characterization of finitely additive interval measures with values in conjugate Banach spaces which can be represented as Henstock-Kurzweil-Gelfand integrals. If the range space has the weak Radon-Nikodým property (WRNP), then we precisely describe when these integrals are in fact Henstock-Kurzweil-Pettis integrals.

Kurzweil--Henstock and Kurzweil--Henstock--Pettis integrability of strongly measurable functions

We study the integrability of Banach valued strongly measurable functions defined on $[0,1]$. In case of functions $f$ given by $\sum _{n=1}^{\infty } x_n\chi _{E_n}$, where $x_n $ belong to a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for the Bochner and for the Pettis integrability of $f$ (cf Musial (1991)). In this paper we give some conditions for the Kurzweil-Henstock and the Kurzweil-Henstock-Pettis integrability of such functions.

A new full descriptive characterization of Denjoy-Perron integral

It is proved that the absolute continuity of the variational measure generated by an additive interval function \(F\) implies the differentiability almost everywhere of the function \(F\) and gives a full descriptive characterization of the Denjoy-Perron integral.

Lineability of non-differentiable Pettis primitives

Let \(X\) be an infinite-dimensional Banach space. In 1995, settling a long outstanding problem of Pettis, Dilworth and Girardi constructed an \(X\)-valued Pettis integrable function on \([0,1]\) whose primitive is nowhere weakly differentiable. Using their technique and some new ideas we show that \(\mathbf{ND}\), the set of strongly measurable Pettis integrable functions with nowhere weakly differentiable primitives, is lineable, i.e., there is an infinite dimensional vector space whose nonzero vectors belong to \(\mathbf{ND}\).

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

AbstractThe 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.