Search

In this paper, we study convergence of nets and sequences of $p$-Bochner, $p$-Dunford and $p$-Pettis integrable functions, $p \in [1, \infty)$, defined on a finite measure space with values in a real Banach space. Applying Holder's inequality, we study some properties of these functions and convergence of their nets and sequences. We introduce the idea of $\delta$-Cauchy nets in terms of which we establish convergence theorems for nets of above types of functions. We see that $\delta$-Cauchyness of a net plays the similar role as uniform integrability does in case of sequences.

In this article, we discuss several aspects of convergence theorems for nets and sequences of Riemann and Riemann-type integrable functions defined on a closed bounded interval in $\mathbb{R}$ with values in a Banach space. We introduce the notions of Riemann $\Delta$-Cauchy nets of functions with its analogous variants and derive some correlations between such kind of nets of functions and equi-Riemann integrability. Moreover, we establish equi-integrability of the pointwise closure of different types of equi-integrable collections of functions. Finally, several related results, e.g., relative compactness of equi-integrable collections of functions with respect to different topologies are studied.

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

For a normal Hilbert $B(H)$-module $X$ gramian Schauder basic $X$-valued measures are considered. Some equivalence conditions are given for an $X$-valued measure to be gramian Schauder basic. As an application gramian uniformly bounded linearly stationary $X$-valued processes are characterized.

Let $X$ be a locally convex Hausdorff space with topological dual $X^*$ and $m$ be a ($\sigma$-additive) $X$-valued vector measure defined on a $\sigma$-algebra. The completeness of the associated $L^1$-space of $m$ is determined by the \emph{closedness} of $m$, a concept introduced by I. Kluvánek in the early 1970's. He characterized the closedness of $m$ via the existence of a certain kind of localizable, $[0,\infty]$- valued measure $\iota$ such that every scalar measure $\langle m,x^*\rangle:E\mto \langle m(E),x^*\rangle$, for $x^*\in X^*$, satisfies $\langle m,x^*\rangle<< \iota$. The construction of $\ia$ relies on the theory of conical measures. Unfortunately, in this generality the characterization is invalid; a counterexample is exhibited. However, by restricting $\iota$ to the class of \emph{Maharam measures} and strengthening the requirement of absolute continuity to the condition that every $\langle m,x^*\rangle$, for $x^*\in X^*$, is \emph{truly continuous } with respect to $\iota$ (a notion investigated by D. Fremlin in connection with the Radon Nikodým Theorem), it is shown that an adequate characterization of the closedness of $m$ is indeed available.

In this paper we study equi-Riemann and equi-Riemann-type integrability of a collection of functions defined on a closed interval of \(\mathbb{R}\) with values in a Banach space. We obtain some properties of such collections and interrelations among them. Moreover we establish equi-integrability of different types of collections of functions. Finally, we obtain relations among equi-Riemann integrability with other properties of a collection of functions.

We introduce the notion of the the weak integral by partitions of unity for functions defined on a \(\sigma\)-finite outer regular quasi Radon measure space \((S,\Sigma,\mathcal{T},\mu)\) into a Banach space \(X\) and discuss its relation with the weak McShane integral which has been introduced by M. Saadoune and R. Sayyad (2014).

We present a natural way to cover an Archimedean directed ordered vector space $E$ by Banach spaces and extend the notion of Bochner integrability to functions with values in $E$. The resulting set of integrable functions is an Archimedean directed ordered vector space and the integral is an order preserving map.

Some limit and Dieudonne-type theorems in the setting of (l)-groups with respect to filter convergence are proved, extending earlier results.

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

The Fubini theorem for the generalized Dobrakov integral in com- plete bornological locally convex topological vector spaces is proven.

We consider integration of functions with values in a partially ordered vector space, and two notions of extension of the space of integrable functions. Applying both extensions to the space of real valued simple functions on a measure space leads to the classical space of integrable functions.

In this paper we define non-archimedean measures and integral operators taking values in a locally convex space. We show the relation between these two concept. We define what we called integral function respect to an integral operator. We give necessary and sufficient condition in order to know when a function is integrable with respect to an integral operator. In the second part, we define a kind of absolutely continuous relation between measures in this context. After that, we formulate a type of Radon--Nikodym Theorem between vector measures and a scalar measures which are absolutely continuous.

Two applications of nets are given. The first is an extension of the Bochner integral to arbitrary locally convex spaces, leading to an integration theory of more general vector valued functions then in the classical approach by Gelfand and Pettis. The second application starts with the observation that an operator on a Hilbert space is trace class if and only if the net of "principal trace minors" converges. The notion of a "determinant class operator" then is defined as one for which the net of determinantal principal minors converges. It is shown that for a normal operator $A$ this condition coincides with $1-A$ being trace class.

Riemann, Riemann-Dunford, Riemann-Pettis and Darboux integrable functions with values in a Banach space and Riemann-Gelfand integrable functions with values in the dual of a Banach space are studied in the light of the work of Graves, Alexiewicz and Orlicz, and Gordon. Various properties of these types of integrals and the interrelation between them are established. The Fundamental Theorem of Integral Calculus for these types of integrals is also studied.

Spaces of scalar functions that are integrable in the sense of Bartle-Dunford-Schwartz integration, with respect to a~convexly bounded vector measure $\mu$, are studied. For instance, under the assumption that the range space $X$ of $\mu$ is sequentially complete, the effect of the Orlicz-Pettis property (with respect to a weaker topology on $X$) on the size of $L^1(\mu)$ is investigated. Some completeness properties of the space $L^1_\bullet(\mu)$ of `scalarly integrable functions' are established for general $X$.

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.

This work introduces microlocal compactness forms (MCFs) as a new tool to study oscillations and concentrations in $\mathrm{L}^p$-bounded sequences of functions. Decisively, MCFs retain information about the location, value distribution, and direction of oscillations and concentrations, thus extending at the same time the theories of (generalized) Young measures and H-measures. In $\mathrm{L}^p$-spaces oscillations and concentrations precisely discriminate between weak and strong compactness, and thus MCFs allow to quantify the difference in compactness. The definition of MCFs involves a Fourier variable, whereby also differential constraints on the functions in the sequence can be investigated easily - a distinct advantage over Young measure theory. Furthermore, pointwise restrictions are reflected in the MCF as well, paving the way for applications to Tartar's framework of compensated compactness; consequently, we establish a new weak-to-strong compactness theorem in a "geometric" way. After developing several aspects of the abstract theory, we consider three applications: For lamination microstructures, the hierarchy of oscillations is reflected in the MCF. The directional information retained in an MCF is harnessed in the relaxation theory for anisotropic integral functionals. Finally, we indicate how the theory pertains to the study of propagation of singularities in certain systems of PDEs. The proofs combine measure theory, Young measures, and harmonic analysis., 59 pages, 3 figures

A new concept of McShane-tightness is introduced to pass from scalar (alias weak) McShane integrability to Pettis integrability. It is used also to derive a scalar McShane version of the Vitali theorem.

Ivan Dobrakov has initiated a theory of non-additive set functions defined on a ring of sets intended to be a non-additive generalization of the theory of finite non-negative countably additive measures. These set functions are now known as the Dobrakov submeasures. In this paper we extend Dobrakov's considerations to vector-valued submeasures defined on a ring of sets. The extension of such submeasures in the sense of Drewnowski is also given., 17 pages

This paper extends the theory of scalar-valued integrable functions with respect to vector measures defined on $δ$-rings to the case of vector-valued tensor integrable functions with respect to vector measures defined on $δ$-rings. This paper also generalizes some results of G. F. Stefánsson for tensor integration theory of vector-valued functions with respect to vector measures defined on $σ$-algebras.

‎Recently‎, ‎Suzuki [Nonlinear Anal‎. ‎71 (2009)‎, ‎no‎. ‎11‎, ‎5313-5317.]‎ ‎published a paper on which Edelstein's fixed theorem was generalized‎. ‎In this manuscript‎, ‎we give some theorems which‎ ‎are the generalization of the fixed theorem of Suzuki's Theorems and thus Edelstein's result [J‎. ‎London Math‎. ‎Soc‎. ‎37 (1962)‎, ‎74-79]‎.

The variational McShane integral for functions taking values in a locally convex space is defined, and it is characterized by means of the p-variation of the indefinite Pettis integral

The aim of this paper is to give some equivalent ways of defining McShane integral for vector valued functions.

Let $f : [a,b] \subseteq \bar{\mathbb{R}}\to \mathcal{E}$ and $\phi : [a,b]\to \mathcal{F}$, where $(\mathcal{E},\mathcal{F},\mathcal{G})$ is a Banach space triple. a) We prove that if $\phi$ is continuous $[c,d ]\to [a,b]$ and $f\circ \psi \cdot d\phi\circ \psi$ is Kurzweil or Henstock variationally integrable, then so is $f\cdot d\phi$ and fulfills the well known change of variable formula. It follows that if $\psi$ is an indefinite Henstock integral and if $f\circ\psi\, \psi' dx $ is K-H integrable, then so is $f dx$ and the change of variable formula applies. b) We produce several versions of the converse of a), that is, we give necessary and sufficient conditions in order that with $\psi$ as above, the integrability of $f\cdot d\phi$ implies that of $f\circ \psi \cdot d\phi\circ\psi$ and the change of variable formula.

A characterization of Banach spaces possessing the Radon–Nikodým 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.

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

Properties of McShane and Kurzweil-Henstock integrable functions taking values in a locally convex space are considered and the relations with other integrals are studied. A convergence theorem for the Kurzweil-Henstock integral is given

The unique local existence is established for the Cauchy problem of the incompressible Navier-Stokes equations with the Coriolis force for a class of initial data nondecreasing at space infinity. The Coriolis operator restricted to divergence free vector fields is a zero order pseudodifferential operator with the skew-symmetric matrix symbol related to the Riesz operator. It leads to the additional term in the Navier-Stokes equations which has real parameter being proportional to the speed of rotation. For initial datum as Fourier preimage of finite Radon measures having no-point mass at the origin we show that the length of existence time-interval of mild solution is independent of the rotation speed.

The Ward and Perron integrals, with respect to abstract derivation bases, for Riesz-space-valued functions are introduced. It is shown that they are equivalent to the respectively defined Henstock-Kurzweil integral.

In this paper properties of Kurzweil-Henstock and Kurzweil-Henstock-Pettis integrals for vector-valued functions are studied. In particular, the absolute integrability for Kurzweil-Henstock integrable functions is characterized and a Kurzweil-Henstock version of the Vitali Theorem for Pettis integrable functions is given.

We give a partial answer to the question in the title by showing that the McShane and Pettis integrals coincide for functions with values in super-reflexive spaces as well as for functions with values in $c_0(\Gamma)$. We also improve an example of Fremlin and Mendoza, according to which these integrals do not coincide in general, by showing that, at least under the Continuum Hypothesis, there is a scalarly negligible function which is not McShane integrable.

In this paper, we define and study the McShane integral of functions mapping a compact interval $I_0$ in $R^m$ into a Banach space $X$. We compare this integral with the Pettis integral and prove, in particular, that the two integrals are equivalent if $X$ is reflexive and the unit ball of the dual $X^*$ satisfies an additional condition (P). This gives additional information on an implicitly stated open problem of R.A. Gordon and on the work of D.H. Fremlin and J. Mendoza.

Recently a connection has been found between the improper Kurzweil-Henstock integral on the real line and the integral over a compact space. In this paper these results are extended to a Pettis-type integral for the case of functions with values in Riesz spaces with ``enough" order continuous functionals.

Using different types of absolute continuity, we characterize additive interval functions which are the primitives of Pettis or strongly McShane integrable functions.

We define an integral of a vector-valued function $f:\Omega\longrightarrow X$ with respect to a bounded countably additive vector-valued measure $\nu:\Sigma\longrightarrow Y$ and investigate its properties. The integral is an element of $X\check{\otimes}Y$, and when $f$ is $\nu$-measurable we show that $f$ is integrable if and only if $\|f\|\in L_{1}(\nu)$. In this case, the indefinite integral of $f$ is of bounded variation if and only if $\|f\|\in L_{1}(|\nu|)$. We also define the integral of a weakly $\nu$-measurable function and show that such a function $f$ satisfies $x^{*}f\in L_{1}(\nu)$ for all $x^{*}\in X^{*}$ and is $|y^{*}\nu|$-Pettis integrable for all $y^{*}\in Y^{*}$.

If $\lambda$ is a vector measure with values in a Banach space and $p > 1$, we consider the space of real functions $L_p(\lambda)$ that are $p$-integrable with respect to $\lambda$. We define two different vector valued dual topologies and we prove several compactness results for the unit ball of $L_p(\lambda)$. We apply these results to obtain new Grothendieck-Pietsch type factorization theorems.

We shall develop the properties of an integral for Banach-valued functions. The formalism is the generalized Riemann integral introduced by Kurzweil \cite{Kur} and Henstock \cite{Hen}. More precisely, the presentation is close to the McShane approach \cite{McS}. Besides its simplicity of presentation, four advantages characterize this theory: %{\leftskip=1cm \item{(i)} the definition can be used for real-valued functions, and can be generalized without modification to general real and complex Banach spaces; \item{(ii)} when a function is integrable its norm is also integrable, and the proof is straightforward from the definition; \item{(iii)} for finite dimension spaces the theory is equivalent to the McShane's theory, which is itself equivalent to the Lebesgue's theory; \item{(iv)} and lastly, for general Banach space, we can prove the equivalence to the Bochner's theory. \par%}

A type of Strassen’s Theorem for measures taking values in the pos-itive cone of a Banach lattice is proved. We generalize a result of A.Hirshberg and R. M. Shortt and formulate a type of Strassen’s Theoremin a topological context via closed sets. 1 Introduction In joint work, M. Marz and R. M. Shortt [10, Theorem 3.7] generalize a versionof the theorem known in probability theory as \Strassen’s Theorem" (see [13],[6], [5, x11.6]) to the context of measures assuming values in a reexive Banachlattice.Continuing the line of inquiry of [12], in [8, Theorem 2] A. Hirshberg andR. M. Shortt prove a result of this type for measures taking values in Banachlattices of a certain type: the KB-spaces. Since reexive )KB )ordercomplete, their result is a generalization of [10, Theorem 3.7].In this paper we give a formulation of Strassen’s Theorem for measurestaking values in Banach lattices with order continuous norm [Theorem 3.10].These spaces occupy a position between the KB-spaces and the order completeBanach lattices (reexive )KB )order continuous norm )order complete),hence our result is a generalization of [8, Theorem 2].We also formulate a type of Strassen’s Theorem in a topological contextvia closed sets [Theorem 3.15].