Henstock-Kurzweil and McShane product integration; Descriptive definitions

The Henstock-Kurzweil and McShane product integrals generalize the notion of the Riemann product integral. We study properties of the corresponding indefinite integrals (i.e. product integrals considered as functions of the upper bound of integration). It is shown that the indefinite McShane product integral of a matrix-valued function A is absolutely continuous. As a consequence we obtain that the McShane product integral of A over [a, b] exists and is invertible if and only if A is Bochner integrable on [a, b]. Supported by grant No. 201/04/0690 of the Grant Agency of the Czech Republic.     

Rm上Banach-值函数的强McShane积分

在本文中,我们定义和研究了I0Rm到Banach空间X中函数的强McShane积分,直接证明了强Mcshane积分与Bochner积分是等价的,McShane积分与强Mcshane积分等价当且仅当Banach空间X有限维. 从而部分地回答了R.A.Gordon的一个公开问题.     

On integration of vector functions with respect to vector measures

We study integration of Banach space-valued functions with respect to Banach space-valued measures. We focus our attention on natural extensions to this setting of the Birkhoff and McShane integrals. The corresponding generalization of the Birkhoff integral was first considered by Dobrakov under the name S*-integral. Our main result states that S*-integrability implies McShane integrability in contexts in which the later notion is definable. We also show that a function is measurable and McShane integrable if and only if it is Dobrakov integrable (i.e. Bartle *-integrable).     

A RIEMANN-TYPE DEFINITON OF THE BOCHNER INTEGRAL

We define the strong McShane integral and prove its equivalence with the Bochner integral In other words,we have given a Riemann-type definition of the Bochner integral.     

On the limits of generalization of the Kolmogorov integral

The generalization of the Kolmogorov integral to functions with values in a Banach space is considered. It is proved that the resulting integral turns out to be essentially more general than the Bochner integral and is exactly equivalent to an integral of McShane type, whose definition requires that the scaling function be measurable.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 258–272.Original Russian Text Copyright © 2005 by A. P. Solodov.This revised version was published online in April 2005 with a corrected issue number.     

On Henstock–Kurzweil and McShane integrals of Banach space-valued functions

This paper deals with the relation between the McShane integral and the Henstock–Kurzweil integral for the functions mapping a compact interval into a Banach space X and some other questions in connection with the McShane integral and the Henstock–Kurzweil integral of Banach space-valued functions. We prove that if a Banach space-valued function f is Henstock–Kurzweil integrable on I₀ and satisfies Property (P), then I₀ can be written as a countable union of closed sets E_n such that f is McShane integrable on each E_n when X contains no copy of c₀. We further give an answer to the Karták's question.     

A characterization of absolutely summing operators by means of McShane integrable functions

Absolutely summing operators between Banach spaces are characterized by means of McShane integrable functions.     

PARTIAL BOCHNER INTEGRABILITY (I)

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The weak McShane integral

We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a σ-finite outer regular quasi Radon measure space (S,Σ, T, µ) into a Banach space X and study its relation with the Pettis integral. In accordance with this new method of integration, the resulting integral can be expressed as a limit of McShane sums with respect to the weak topology. It is shown that a function f from S into X is weakly McShane integrable on each measurable subset of S if and only if it is Pettis and weakly McShane integrable on S. On the other hand, we prove that if an X-valued function is weakly McShane integrable on S, then it is Pettis integrable on each member of an increasing sequence (S _l ) _l?1 of measurable sets of finite measure with union S. For weakly sequentially complete spaces or for spaces that do not contain a copy of c ₀, a weakly McShane integrable function on S is always Pettis integrable. A class of functions that are weakly McShane integrable on S but not Pettis integrable is included.     

Absolutely summing operators and integration of vector-valued functions

Let (Ω,Σ,μ) be a complete probability space and an absolutely summing operator between Banach spaces. We prove that for each Dunford integrable (i.e., scalarly integrable) function the composition u○f is scalarly equivalent to a Bochner integrable function. Such a composition is shown to be Bochner integrable in several cases, for instance, when f is properly measurable, Birkhoff integrable or McShane integrable, as well as when X is a subspace of an Asplund generated space or a subspace of a weakly Lindelöf space of the form C(K). We also study the continuity of the composition operator f?u○f. Some other applications are given.     

On the Pettis Integral of Closed Valued Multifunctions

In this work, the study of Pettis integrability for multifunctions (alias set-valued maps), whose values are allowed to be unbounded, is initiated. For this purpose, two notions of Pettis integrability, and of Pettis integral, are considered and compared. The first notion is similar to that of the weak integral, already known for vector-valued functions, and is defined via support functions. The second notion resembles the classical Aumann definition using integrable selections, but it involves the Pettis integrable selections rather than the Bochner integrable ones. The above two integrals are shown to coincide in a quite general setting. Several criteria for a multifunction to be Pettis integrable (in one sense or the other) are proved. On the other hand, due to the possibility of infinite values for the support functions, we are led to introduce a more general notion of scalar integrability involving the negative part of these functions. We compare the scalar integrability of a multifunction with that of its measurable selections. We also provide some new results concerning multifunctions with bounded values and/or new proofs of already existing ones. Examples are included to illustrate the results and to introduce open problems.     

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.     

Nonstandard Representation Theory of Standard Operators Defined on the Space of Bochner Integrable Functions

We introduce and study several nonstandard representations of Banach‐valued operators defined on the space of Bochner integrable functions. They will be less restrictive than the usual standard representation. In particular, without any hypothesis, we shall find a representation whose kernel belongs to a space of “extended Bochner integrable functions”, introduced by Zimmer by using Loeb measures.     

On the equivalence of McShane and Pettis integrability in non-separable Banach spaces

We show that McShane and Pettis integrability coincide for functions , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelöf determined Banach space X, a scalarly null (hence Pettis integrable) function and an absolutely summing operator u from X to another Banach space Y such that the composition is not Bochner integrable; in particular, h is not McShane integrable.     

Henstock积分问题的一个注记

本文证明了如果X是不含c0的Banach空间，f是定义在区间I0包含R^m上取值于Panach空间X的函数，并且，在I0上Henstock可积，则总存在I0的一个非退化子区间J，使得f在J上McShane可积，从而对Kartak的一个问题作出了肯定的回答．     

Conditions for holomorphic extendability of functions from borders of domains

We consider extensions of integrable functions by terms of the Bochner–Martinelli integral.     

Trace formulas for nuclear operators in spaces of Bochner integrable functions

The paper is devoted to trace formulas for nuclear operators in spaces of Bochner integrable functions. We characterise nuclearity for integral operators on such spaces and develop a trace formula for general kernels applying vector-valued maximal functions.     

On the equivalence of McShane and Pettis integrability in non-separable Banach spaces

We show that McShane and Pettis integrability coincide for functions , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelöf determined Banach space X, a scalarly null (hence Pettis integrable) function and an absolutely summing operator u from X to another Banach space Y such that the composition is not Bochner integrable; in particular, h is not McShane integrable.     

On Lineability in Vector Integration

We prove the existence of infinite-dimensional linear spaces of Banach space-valued functions whose non-zero elements witness that two given notions of integrability are different: Bochner, Birkhoff, McShane, Pettis and Dunford integrability are considered.     

The McShane integral in weakly compactly generated spaces

Di Piazza and Preiss asked whether every Pettis integrable function defined on [0,1] and taking values in a weakly compactly generated Banach space is McShane integrable. In this paper we answer this question in the negative. Moreover, we give a counterexample where the target Banach space is reflexive.