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 the Strong McShane Integral of Functions with Values in a Banach Space

The classical Bochner integral is compared with the McShane concept of integration based on Riemann type integral sums. It turns out that the Bochner integrable functions form a proper subclass of the set of functions which are McShane integrable provided the Banach space to which the values of functions belong is infinite-dimensional. The Bochner integrable functions are characterized by using gauge techniques. The situation is different in the case of finite-dimensional valued vector functions.     

Some Full Descriptive Characterizations of the Henstock-Kurzweil Integral in the Euclidean Space

Using generalized absolute continuity, we characterize additive interval functions which are indefinite Henstock-Kurzweil integrals in the Euclidean space.     

Henstock-Kurzweil delta and nabla integrals

We will study the Henstock-Kurzweil delta and nabla integrals, which generalize the Henstock-Kurzweil integral. Many properties of these integrals will be obtained. These results will enable time scale researchers to study more general dynamic equations. The Henstock-Kurzweil delta (nabla) integral contains the Riemann delta (nabla) and Lebesque delta (nabla) integrals as special cases.     

R~n中Banach值函数的Mcshane积分(I)(英文)

R．AGordon在［1］中定义了从R1到Banach空间抽象函数的McShane积分，证明了当X不含C0时，如果f在[a，b]上McShanef可积，则在[a，b]上Petits　可积．在这篇文章中，我们定义了从Rn到Banaach空间抽象函数的Mcshane积分，证明了fMcShane可积，则f是Pattis可积．于是我们推广了[1]的结果．     

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 measure-theoretic characterization of the Henstock-Kurzweil integral revisited

In this paper we show that the measure generated by the indefinite Henstock-Kurzweil integral is F _σδ regular. As a result, we give a shorter proof of the measure-theoretic characterization of the Henstock-Kurzweil integral.     

On the Henstock-Kurzweil Integral for Riesz-Space-Valued Functions Defined on Unbounded Intervals

In this paper we introduce and investigate a Henstock-Kurzweil-type integral for Riesz-space-valued functions defined on (not necessarily bounded) subintervals of the extended real line. We prove some basic properties, among them the fact that our integral contains under suitable hypothesis the generalized Riemann integral and that every simple function which vanishes outside of a set of finite Lebesgue measure is integrable according to our definition, and in this case our integral coincides with the usual one.     

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.     

Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral II

This paper is a continuation of the paper [T.Y. Lee, Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral, J. Math. Anal. Appl. 298 (2004) 677-692], in which we proved several Fubini-Tonelli type theorems for the Henstock-Kurzweil integral. Let f be Henstock-Kurzweil integrable on a compact interval . For a given compact interval , set

     

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.     

Estimates of the Remainder in Taylor's Theorem Using the Henstock-Kurzweil Integral

When a real-valued function of one variable is approximated by its nth degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue p-norms in cases where f ⁽ⁿ⁾ or f ⁽ⁿ⁺¹⁾ are Henstock-Kurzweil integrable. When the only assumption is that f ⁽ⁿ⁾ is Henstock-Kurzweil integrable then a modified form of the nth degree Taylor polynomial is used. When the only assumption is that f ⁽ⁿ⁾ ∈ C ⁰ then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1. Research partially supported by the Natural Sciences and Engineering Research Council of Canada. An adjunct appointment in the Department of Mathematical and Statistical Sciences, University of Alberta, made valuable library and computer resources available.     

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.     

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.     

The Distributional Henstock-Kurzweil Integral and Applications: A Survey

This study presents a summary of the current state of research on the distributional Henstock-Kurzweil integral. Basic properties such as integration by parts, Hake theorem, inner product, Hölder inequality, second mean value theorem, orderings, Banach lattice, convergence theorems, fixed point theorems, are shown. This study also summarizes its applications in integral and differential equations.     

Denjoy integral and Henstock-Kurzweil integral in vector lattices,I

In this paper we define the derivative and the Denjoy integral of mappings from a vector lattice to a complete vector lattice and show the fundamental theorem of calculus.      

Henstock and McShane integrals for Banach-valued functions

We solve the problem of the equivalence of theH L-integral and the Henstock integral in Banach spaces. Namely, we prove that the Saks-Henstock lemma is valid in a Banach space if and only if it is finite-dimensional. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 860–870, June, 1999.     

Denjoy integral and Henstock-Kurzweil integral in vector lattices,II

In a previous paper we defined a Denjoy integral for mappings from a vector lattice to a complete vector lattice. In this paper we define a Henstock-Kurzweil integral for mappings from a vector lattice to a complete vector lattice and consider the relation between these two integrals.      

Differential and integral equations with Henstock-Kurzweil integrable functions

In this paper we apply fixed point theorems for increasing mappings in ordered normed spaces to prove existence and comparison results for solutions of discontinuous functional differential and integral equations containing Henstock-Kurzweil integrable functions.     

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.