Anna Rita Sambucini

Vitali-type theorems for filter convergence related to vector lattice-valued modulars and applications to stochastic processes☆

A Vitali-type theorem for vector lattice-valued modulars with respect to filter convergence is proved. Some applications are given to modular convergence theorems for moment operators in the vector lattice setting, and also for the Brownian motion and related stochastic processes.

Some inequalities in classical Spaces with mixed norms

We consider some inequalities in such classical Banach Function Spaces as Lorentz, Marcinkiewicz, and Orlicz spaces. Our aim is to explore connections between the norm of a function of two variables on the product space and the mixed norm of the same function, where mixed norm is calculated in function spaces on coordinate spaces, first in one variable, then in the other. This issue is motivated by various problems of functional analysis and theory of functions. We will currently mention just geometry of spaces of vector-valued functions and embedding theorems for Sobolev and Besov spaces generated by metrics which differ from Lp. Our main results are actually counterexamples for Lorentz spaces versus the natural intuition that arises from the easier case of Orlicz spaces (Section 2). In the Appendix we give a proof for the Kolmogorov–Nagumo theorem on change of order of mixed norm calculation in its most general form. This result shows that Lp is the only space where it is possible to change this order.

A NOTE ON COMPARISON BETWEEN BIRKHOFF AND MCSHANE-TYPE INTEGRALS FOR MULTIFUNCTIONS

Here we present some comparison results between Birkho and McShane multivalued integration.

The Finitely Additive Integral of Multifunctions with Closed and Convex Values

We investigate integration with respect to a finitely additive measure of integrands with closed, convex values and we obtain a closedness result for the Aumann integral.

Order-type Henstock and McShane integrals in banach lattice setting

Henstock-type integrals are studied for functions defined in a compact metric space T endowed with a regular σ-addltive measure μ, and taking values in a Banach lattice X. In particular, the space [0,1] with the usual Lebesgue measure is considered. The norm- and the order-type integral are compared and interesting results are obtained when X is an L-space. 2010 AMS Mathematics Subject Classification: 28B20, 46G10.

An Extension of the Birkhoff Integrability for Multifunctions

A comparison between a set-valued Gould type and simple Birkhoff integrals of bf(X)-valued multifunctions with respect to a non-negative set function is given. Relationships among them and Mc Shane multivalued integrability is given under suitable assumptions.

L^p spaces in vector lattices and applications

Abstract Lp spaces are investigated for vector lattice-valued functions, with respect to filter convergence. As applications, some classical inequalities are extended to the vector lattice context, and some properties of the Brownian motion and the Brownian bridge are studied, to solve some stochastic differential equations.

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.

Comparison between different types of abstract integrals in Riesz spaces

A comparison among different types of integral in Riesz spaces is given.

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

The aim of this paper is to study relationships among “gauge integrals” (Henstock, McShane, Birkhoff) and Pettis integral of multifunctions whose values are weakly compact and convex subsets of a general Banach space, not necessarily separable. For this purpose, we prove the existence of variationally Henstock integrable selections for variationally Henstock integrable multifunctions. Using this and other known results concerning the existence of selections integrable in the same sense as the corresponding multifunctions, we obtain three decomposition theorems (Theorems 3.2, 4.2, 5.3). As applications of such decompositions, we deduce characterizations of Henstock (Theorem 3.3) and