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The aim of this paper is to investigate different types of multi-integrals of finite variation and to obtain decomposition results., 9 pages

Integral properties of multifunctions determined by vector valued functions are presented. Such multifunctions quite often serve as examples and counterexamples. In particular it can be observed that the properties of being integrable in the sense of Bochner, McShane or Birkhoff can be transferred to the generated multifunction while Henstock integrability does not guarantee it., Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1812.00597

A kind of Pettis integral representation for a Banach valued It\^o process is given and its drift term is modified using a Girsanov Theorem., Comment: 12 pages

It has been proven in previous papers that each Henstock-Kurzweil-Pettis integrable multifunction with weakly compact values can be represented as a sum of one of its selections and a Pettis integrable multifunction. We prove here that if the initial multifunction is also Bochner measurable and has absolutely continuous variational measure of its integral, then it is a sum of a strongly measurable selection and of a variationally Henstock integrable multifunction that is also Birkhoff integrable., Comment: 15 pages

The aim of this paper is to study relationships among "gauge integrals" (Henstock, Mc Shane, 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. As applications of such decompositions, we deduce characterizations of Henstock and ${\mathcal H}$ integrable multifunctions, together with an extension of a well-known theorem of Fremlin., 13 pages

We study Riemann-Lebesgue integrability of a vector function relative to an arbitrary non-negative set function. We obtain some classical integral properties. Results regarding the continuity properties of the integral and relationships among Riemann-Lebesgue, Birkhoff simple and Gould integrabilities are also established., Comment: 16 pages

Some integration techniques for real-valued functions with respect to vector measures with values in Banach spaces (and viceversa) are investigated in order to establish abstract versions of classical theorems of Probability and Stochastic Processes. In particular the Girsanov Theorem is extended and used with the treated methods., 15 pages

In this paper we present some results concerning Gould integrability of vector functions with respect to a monotone measure on finitely purely atomic measure spaces. As an application a Radon-Nikodym theorem in this setting is obtained., Comment: 17 pages

A vector-valued version of the Girsanov theorem is presented, for a scalar process with respect to a Banach-valued measure. Previously, a short discussion about the Birkhoff-type integration is outlined, as for example integration by substitution, in order to fix the measure-theoretic tools needed for the main result, Theorem 6, where a martingale equivalent to the underlying vector probability has been obtained in order to represent the modified process as a martingale with the same marginals as the original one.

A comparison between a set-valued Gould type and simple Birkhoff integrals of $bf(X)$-valued multifunctions with respect to a non-negative set functionis given. Relationships among them and Mc Shane multivalued integrability is given under suitable assumptions., Comment: 24 pages

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 operatorsin the vector lattice setting, and also for the Brownian motion and related stochastic processes., 19 pages, 2 figures

Filter convergence of vector lattice-valued measures is considered, in order to deduce theorems of convergence for their decompositions. First the $\sigma$-additive case is studied, without particular assumptions on the filter; later the finitely additive case is faced, first assuming uniform $s$-boundedness (without restrictions on the filter), then relaxing this condition but imposing stronger properties on the filter. In order to obtain the last results, a Schur-type convergence theorem is used., Comment: 18 pages

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., 21 pages

$L^p$ 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., Comment: 21 pages, 2 figures

Some new results about uniform (s)-boundedness for regular (l)-group-valued set functions are given.

We apply the theory of Differential and Integral Calculus in Riesz Spaces introduced in [1] and [4] to investigate some properties of the g-calculus and to solve some types of differential, functional and stochastic equations.

A convergence in Riesz spaces is given axiomatically. A Bochner-type integral for Riesz space-valued functions is introduced and some Vitali and Lebesgue dominated convergence theorems are proved. Some properties and examples are investigated.

Brownian motions, martingales, and Wiener processes are introduced and studied for set valued functions taking values in the subfamily of compact convex subsets of arbitrary Banach space $X$. The present paper is an application of one the paper of the second author in which an embedding result is obtained which considers also the ordered structure of $ck(X)$ and f-algebras., 15 pages, 1 figure

We study Henstock-type integrals for functions defined in a compact metric space $T$ endowed with a regular $\sigma$-additive measure $\mu$, and taking values in a Banach lattice $X$. In particular, the space $[0,1]$ with the usual Lebesgue measure is considered., Comment: 5 pages

In the theory of quasi-measures (i.e., finitely additive measures) some results have been found ensuring that, under suitable hypotheses, there exists a Radon-Nikodym derivative for pairs of quasi-measures m = (p, v): see for instance [6, 81. More recently, Greco in [S] gave a necessary and sufficient condition for the Radon-Nikodym representation of a pair of monotone set functions: Greco’s theorem becomes especially meaningful when the involved set functions are finitely additive. Moreover, if the quasimeasures are continuous, Greco’s characterization makes it possible to closely relate the Radon-Nikodym derivative to the geometric properties of the range R(m) of the quasi-measure m = (p, v). So, in view of a further Radon-Nikodym theorem, one may find most helpful a recent result obtained in [7] (see Lemma 1.4 below): that is, when p and v are continuous, if R(m) is closed, then it satisfies the “h.0.b.” (hereditarily overlapping boundary) property. More precisely, if m(A)caR(m) for some set A, then the boundaries JR(m) and LTR(m,) partially overlap, where m,., is the quasi-measure defined as m,(B) = m(A n B) VB. In this work, using the above-mentioned results, we show that when p is continuous, v 4 p, and R(m) is closed, a Radon-Nikodym representation holds. The closedness hypothesis on R(m) is not necessary. Indeed, more generally we give a necessary and sufficient condition, in terms of the geometric properties of R(m), for the existence of a Radon-Nikodym derivative. Last we show that every planar set, satisfying those geometric conditions, is the range of some continuous quasi-measure m, whose components admit a Radon-Nikodym representation. 9 0001~8708/92 $9.00

A particular notion of limit is introduced, for Riesz space-valued functions. The definition depends on certain ideals of subsets of the domain. It is shown that, according with our definition, every bounded function with values in a Dedekind complete Riesz space admits limit with respect to any maximal ideal.

Some convergence theorems have been obtained for the $GH_k$ integral for functions defined in abstract topological spaces and with values in Riesz spaces.

This chapter describes the various aspects of Radon–Nikodym theorems. A general Radon–Nikodym theorem for nonnegative finitely additive scalar measures is presented in the chapter from which the basic ideas of further results can be drawn. The conditions that permit even unbounded measures to have Radon–Nikodym derivatives are examined in the chapter. The results concerning Banach spaces possessing the so-called “Radon–Nikodym Property” (RNP) are presented in the chapter. The geometric concepts become essential tools for describing Banach spaces. The finitely additive measures taking values in Banach spaces and to the research of weaker types of derivatives are elaborated in the chapter. To find a characterization of the Radon–Nikodym property, the concept of completeness is required. A complex measure or function can be studied investigating separately its real and imaginary parts. It is proved in the chapter that convergence in measure implies convergence for some subsequence; hence, a strongly measurable function is essentially valuedseparably. It is found that the problems concerning the existence of a Radon–Nikodym derivative are harder when one allows also finitely additive measures into consideration. Some results concerning Radon–Nikodym derivatives for Sugeno integral are also presented in the chapter.

We find a decomposition of the type of Sobczyk-Hammer for measures with values in $l$-groups, and also deduce some convergence theorems for such decompositions. Our procedure is based on some theorems of the type of Vitali-Hahn-Saks, and on the so-called Stone extension method.

Some versions of Dieudonn e theorems are given for set functions, not necessarily positive, taking values in Dedekind complete (l)-groups, relatively to the \(D)-convergence".

Si studiano funzioni d'intervallo, in ambienti astratti e in spazi di misura. Attraverso l'assoluta continuita, si ottengono proprieta di quasi additivita e quasi sub-additivita in una varieta di casi. Si stabilisce infine un teorema di semicontinuita inferiore per l'integrale di Burkill-Cesari.

Absolute continuity, s-boundedness, and extensions are studied, in the context of the so-called RD -convergence, for set functions taking values in Dedekind complete l-groups. Subsequently, we obtain results of uniform s-boundedness for RD -convergent sequences of measures (Vitali–Hahn–Saks–Nikodým theorem) and deduce a Schur-type theorem for measures defined on P ( N *).

We state some approximation theorems for the multiple integral of Calculus of Variations. They are formulated in a very general setting, which includes some classical results on the area of a surface, reached by C. Vinti.

Integral properties of multifunctions with closed convex values are studied. In this more general framework not all the tools and the technique used for weakly compact convex valued multifunctions work. We pay particular attention to the "positive multifunctions". Among them an investigation of multifunctions determined by vector-valued functions is presented. Finally, decomposition results are obtained for scalarly and gauge-defined integrals of multifunctions and a full description of McShane integrability in terms of Henstock and Pettis integrability is given., Comment: 20 pages