巴拿赫格上的序型Henstock和McShane积分

巴拿赫格关于点态非原子测度的Henstock多值可积性

Henstock multivalued integrability in Banach lattices with respect to pointwise non atomic measures

本文研究了巴拿赫格$X$的弱紧凸子集族中的集值函数的Henstock型积分。处理多值情形的主要工具是C. C. A. Labuschagne，A. L.平查克角2007年的J. van Alten。这样，模积分和序积分就化为取值于$M$ -空间的单值函数的模积分和序积分，并对Di Piazza和Musial最近的两篇论文中基于可积选择的存在性的分解结果给出了新的证明.

Countable additivity of Henstock-Dunford Integral and Orlicz Space

Given a real Banach space $\mathcal{X}$ and probability space $(Ω, Σ, μ)$ we characterize the countable additivity of Henstock-Dunford integral for Henstock integrable function taking values in $X$ as those weakly measurable function $ g: Ω\to \mathcal{X} $ for which $\{y^*g~: y^* \in B_\mathcal{X}^* \} $ is relatively weakly compact in some separable Orlicz space $ \mathcal{L}^{\overlineϕ}(μ) .$ We find relatively weakly compact in some Orlicz space with Henstock-Gel'fand integral.

Gauge integrals and selections of weakly compact valued multifunctions

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.

Multifunctions determined by integrable functions

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.

Full characterizations of the variational McShane Integral on $m$-dimensional compact intervals

In this paper we consider the additive interval functions defined on the family $\mathcal{I}_{m}$ of all non-degenerate closed subintervals of the cubic interval $C_{m} = [0,1]^{m}$ in the $m$-dimensional Euclidean space $\mathbb{R}^{m}$ and taking values in a Banach space $X$. We give necessary and sufficient conditions for an additive interval function $F : \mathcal{I}_{m} \to X$ to be the primitive of a variational McShane (or strong McShane) integrable function $f : C_{m} \to X$ in terms of the convex cubic average range of $F$.

Integration of multifunctions with closed convex values in arbitrary Banach spaces

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.

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

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 decompos…

弱紧生成空间中的McShane积分

The McShane integral in weakly compactly generated spaces

迪Piazza和Preiss问是否每一个Pettis可积函数定义在[0，1]和采取的价值观在一个弱紧生成的巴拿赫空间是McShane可积。在本文中，我们回答这个问题是否定的。

巴拿赫格中几种范数和序规范积分的比较

Comparison between some norm and order gauge integrals in Banach lattices

考虑了函数在Radon测度空间中的Henstock型积分，其取值于巴拿赫格X，并考虑了空间的范数和序结构.得到了一些结果，突出了它们之间的差异，特别是在L-空间值函数的情况下。

Open problems in Banach spaces and measure theory

We collect several open questions in Banach spaces, mostly related to measure theoretic aspects of the theory. The problems are divided into five categories: miscellaneous problems in Banach spaces (non-separable $L^p$ spaces, compactness in Banach spaces, $w^*$-null sequences in dual spaces), measurability in Banach spaces (Baire and Borel $σ$-algebras, measurable selectors), vector integration (Riemann, Pettis and McShane integrals), vector measures (range and associated $L^1$ spaces) and Lebesgue-Bochner spaces (topological and structural properties, scalar convergence).

Kuelbs-Steadman spaces on Separable Banach spaces

The purpose of this paper is to construct a new class of separable Banach spaces $\K^p[\mathbb{B}], \; 1\leq p \leq \infty$. Each of these spaces contain the $ \mcL^p[\mathbb{B}] $ spaces, as well as the space $\mfM[\R^\iy]$, of finitely additive measures as dense continuous compact embeddings. These spaces are of interest because they also contain the Henstock-Kurzweil integrable functions on $\mathbb{B}$. Finally, we offer a interesting approach to the Fourier transform on $\K^p[\mathbb{B}].$

关于向量值函数的积分

On the integration of vector-valued functions

我们讨论了McShane，Pettis，Talagrand和Bochner积分之间的关系。基于勒贝格积分的各种可能的构造，已经引入了大量不同的Banach空间值函数的积分方法。当值域空间是可分的（或具有w^*-可分对偶）时，它们通常相当紧密地在一起运行，而对于一般值域空间，它们或多或少地急剧发散。由[Go]描述的McShane积分是从[McS]的“规范极限”积分导出的。在这里我们给出关于它和上面列出的其他三个积分的正的和负的结果。

任意 Banach 空间中可积多值函数的选择

Selections of integrable multifunctions in arbitrary Banach spaces

该研究在任意（不必可分的）Banach 空间中考察多值函数的可积选择存在性，并用以将多值函数表示为性质更好的多值函数之平移。

$C(K)-$空间上的格 Lipschitz 算子

Lattice Lipschitz operators on $C(K)-$space

本文研究了连续函数空间上的格 Lipschitz 算子，探讨了其函数界、表示定理与对偶空间，并提供了 McShane-Whitney 扩张定理以服务于人工智能应用。

Banach spaces for the Schwartz distributions

This paper is a survey of a new family of Banach spaces ${KS}^2$ and $SD^2$ that provide the same structure for the Henstock-Kurzweil (HK) integrable functions as the $L^p$ spaces provide for the Lebesgue integrable functions. These spaces also contain the wide sense Denjoy integrable functions. They were first use to provide the foundations for the Feynman formulation of quantum mechanics. It has recently been observed that these spaces contain the test functions $\mathcal{D}$ as a continuous dense embedding. Thus, by the Hahn-Banach theorem, $\mathcal{D}' \subset \mathcal{B}'$. A new family …

调节本原积分

The regulated primitive integral

如果一个函数在实数直线上的每一点上都有一个左极限和一个右极限，则该函数被称为调节函数。如果$f$是一个Schwartz分布在实数线上，使得$f=F'$（分布或弱导数）对于一个调节函数$F$，那么$f$的调节原始积分是$\int_{(a,b)}f=F(b-)-F(a+)$，对于其他类型的区间有类似的定义。可积分布空间是一个巴拿赫空间，在Alexiewicz范数下是一个巴拿赫格。它包含空间的Lebesgue和Henstock-Kurzweil可积函数作为连续嵌入。它是符号Radon测度空间在Alexiewicz范数下的完备化。有界变差的函数形成对偶空间和乘子空间。可积分布是有界变差函数上的模。证明了分部积分法、变量变换、赫尔德不等式、泰勒定理和收敛性定理等性质。

有界Lipschitz函数空间中单位球的极端点的规范集与稠密集

Norming and dense sets of extreme points of the unit ball in spaces of bounded Lipschitz functions

本文在有界Lipschitz函数空间中，通过McShane扩展算子构造规范极端函数子集，证明其在单位球极端点中弱星稠密，为Dudley范数的计算开辟了新途径。

连续本原积分的卷积

Convolutions with the continuous primitive integral

如果$F$是一个连续的函数在实直线上和$f=F'$是它的分布导数，那么连续的原始积分分布$f$是$\int_a^bf=F(b)-F(a)$。该积分包含Lebesgue、Henstock-Kurzweil和宽Denjoy积分。在Alexiewicz范数下，可积分布空间是一个Banach空间。我们定义了卷积$f\ast g(x)=\intinf f(x-y)g(y) dy$，$f$是可积分布，$g$是有界变差函数或$L^1$函数。卷积的基本性质是：交换性，结合性，平移交换。对于有界变差的$g$，$f\ast g$是一致连续的，我们有估计$\|f\ast g\|_\infty\leq \|f\|\|g\|_\bv$，其中$\|f\|=\sup_I|\int_If|$是Alexiewicz范数。该上确界在所有区间$I\subset\R$上取。当$g\in L^1$时，估计值为$\|f\ast g\|\leq \|f\|\|g\|_1$。对卷积的微分和积分有了一些结果。证明了连续型本原积分的一类Fubini定理。

用格Lipschitz算子逼近几乎对角的非线性映射

Approximation of almost diagonal non-linear maps by lattice Lipschitz operators

本文提出利用格Lipschitz算子逼近几乎对角非线性映射的方法，基于逼近性质与误差界结合McShane-Whitney扩展映射插值，并给出误差公式与示例。