The Pettis integral for multi-valued functions via single-valued ones

We study the Pettis integral for multi-functions defined on a complete probability space (Ω,Σ,μ) with values into the family cwk(X) of all convex weakly compact non-empty subsets of a separable Banach space X. From the notion of Pettis integrability for such an F studied in the literature one readily infers that if we embed cwk(X) into ?_∞(BX^∗) by means of the mapping defined by j(C)(x^∗)=sup(x^∗(C)), then j○F is integrable with respect to a norming subset of B?_∞^∗(BX^∗). A natural question arises: When is j○F Pettis integrable? In this paper we answer this question by proving that the Pettis integrability of any cwk(X)-valued function F is equivalent to the Pettis integrability of j○F if and only if X has the Schur property that is shown to be equivalent to the fact that cwk(X) is separable when endowed with the Hausdorff distance. We complete the paper with some sufficient conditions (involving stability in Talagrand's sense) that ensure the Pettis integrability of j○F for a given Pettis integrable cwk(X)-valued function F.     

Convergence theorems for the Birkhoff integral

We give sufficient conditions for the interchange of the operations of limit and the Birkhoff integral for a sequence (f _n ) of functions from a measure space to a Banach space. In one result the equi-integrability of f _n ’s is involved and we assume f _n → f almost everywhere. The other result resembles the Lebesgue dominated convergence theorem where the almost uniform convergence of (f _n ) to f is assumed.     

On the existence of Pettis integrable functions which are not Birkhoff integrable

Let be a weakly Lindelöf determined Banach space. We prove that if is non-separable, then there exist a complete probability space and a bounded Pettis integrable function that is not Birkhoff integrable; when the density character of is greater than or equal to the continuum, then is defined on with the Lebesgue measure. Moreover, in the particular case (the cardinality of being greater than or equal to the continuum) the function can be taken as the pointwise limit of a uniformly bounded sequence of Birkhoff integrable functions, showing that the analogue of Lebesgue's dominated convergence theorem for the Birkhoff integral does not hold in general.

     

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.     

向量值函数的Mcshane积分

本文讨论了向量值函数的Ｍｃｓｈａｎｅ积分的性质．例如收敛定理和原函数的性质等．     

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 Birkhoff integrability for scalar functions and vector measures

The Bartle–Dunford–Schwartz integral for scalar functions with respect to vector measures is characterized by means of Riemann-type sums based on partitions of the domain into countably many measurable sets. In this setting, two natural notions of integrability (Birkhoff integrability and Kolmogoroff integrability) turn out to be equivalent to Bartle–Dunford–Schwartz integrability. A. Fernández, F. Mayoral and F. Naranjo were supported by MEC and FEDER (project MTM2006–11690–C02–02) and La Junta de Andalucía. J. Rodríguez was supported by MEC and FEDER (project MTM2005-08379), Fundación Séneca (project 00690/PI/04) and the Juan de la Cierva Programme (MEC and FSE).     

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.     

奇异积分的三次Birkhoff型插值样条逼近

本文采用三次Birkhoff型插值样条讨论任意光滑弧上的奇异积分T_w(f:x,r)=∫_p(w(t)f(t))/(t-x)dt的逼近,在f(t)∈D_1,权函数w(t)∈D_1.分划序列拟一致的条件下,证明了其一致收敛性.     

Riemann type integrals for functions taking values in a locally convex space

The McShane and Kurzweil-Henstock integrals for functions taking values in a locally convex space are defined and the relations with other integrals are studied. A characterization of locally convex spaces in which Henstock Lemma holds is given.     

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.     

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.     

Measurable selectors and set-valued Pettis integral in non-separable Banach spaces

Kuratowski and Ryll-Nardzewski's theorem about the existence of measurable selectors for multi-functions is one of the keystones for the study of set-valued integration; one of the drawbacks of this result is that separability is always required for the range space. In this paper we study Pettis integrability for multi-functions and we obtain a Kuratowski and Ryll-Nardzewski's type selection theorem without the requirement of separability for the range space. Being more precise, we show that any Pettis integrable multi-function F:Ω→cwk(X) defined in a complete finite measure space (Ω,Σ,μ) with values in the family cwk(X) of all non-empty convex weakly compact subsets of a general (non-necessarily separable) Banach space X always admits Pettis integrable selectors and that, moreover, for each A∈Σ the Pettis integral coincides with the closure of the set of integrals over A of all Pettis integrable selectors of F. As a consequence we prove that if X is reflexive then every scalarly measurable multi-function F:Ω→cwk(X) admits scalarly measurable selectors; the latter is also proved when (X^∗,w^∗) is angelic and has density character at most ω₁. In each of these two situations the Pettis integrability of a multi-function F:Ω→cwk(X) is equivalent to the uniform integrability of the family . Results about norm-Borel measurable selectors for multi-functions satisfying stronger measurability properties but without the classical requirement of the range Banach space being separable are also obtained.     

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 sharp estimate for the rate of convergence in mean of Birkhoff sums for some classes of periodic differentiable functions

For a badly approximable vector α, we obtain a sharp estimate for the rate of convergence in the space L _p (0 < p < ∞) of the Birkhoff means $\frac{1}{n}\sum\nolimits_{s = 0}^{n = 1} {f(x + s\alpha )} $ for an absolutely continuous periodic function f and for functions in spaces of Bessel potentials.     

Birkhoff interpolation on non-uniformly distributed roots of unity

This paper studies some cases of (0,m)-interpolation on non-uniformly distributed roots of unity that were not covered before. The interpolation problem uses as nodes the zeros of (z ^k +1)(z ³–1) with k=3n+1, 3n+2. Proof of the regularity is more intricate than when k is divisible by 3, the case included in a previous paper by the authors. The interpolation problem appears to be regular for mk+3, a result that is in tune with the case k=3n mentioned before. However, it is necessary to treat the full general 18×18 linear system. For small values of m the determinant is calculated explicitly using MAPLE V, Release 5.     

Approximating the integral of a multifunction

Given a weakly converging sequence of measures, we study the convergence of the corresponding integrals of a continuous unbounded multifunction. We also study the implication of these results to variational problems, and provide further approximating results for the integral of a multifunction, involving both truncation of the multifunction and measure approximation.     

流的Birkhoff中心及Ω稳定性条件

首先对紧度量空间上的连续流论证了滤子的存在性与无环性的关系,并给出了Birkhoff中心是非游荡集的一个充分条件;然后对流形上的C1流证明了:Birkhoff中心双曲+无环条件公理A+无环条件,因而它是Ω稳定的.     

Birkhoff Centre of a Poset

In this paper, it is proved that the Boolean centre of a semigroup S with sufficiently many commuting idempotents is isomorphic to the inverse limit of the directed family of Birkhoff centres (or Boolean centres) of a class of bounded semigroups. The Birkhoff centre is defined for any poset and proved that it is a relatively complemented distributive lattice whenever it is nonempty. It is observed that for a semilattice S, the Birkhoff centres as a semigroup and as a poset coincide. Also it is observed that for a Lattice (L, , ), the Birkhoff centres of the semilattices (L, ) and (L, ) coincide with the Birkhoff centre of L. Finally it is proved that for a lattice (L, , ), the Boolean centres of the semilattices (L, ) and (L, ) coincide with the Boolean centre of L.AMS Subject classification (1991): 06A12, 20M15     

A quadratic-phase integral operator for sets of generalized integrable functions

In this paper, we aim to discuss the classical theory of the quadratic-phase integral operator on sets of integrable Boehmians. We provide delta sequences and derive convolution theorems by using certain convolution products of weight functions of exponential type. Meanwhile, we make a free use of the delta sequences and the convolution theorem to derive the prerequisite axioms, which essentially establish the Boehmian spaces of the generalized quadratic-phase integral operator. Further, we nominate two continuous embeddings between the integrable set of functions and the integrable set of Boehmians. Furthermore, we introduce the definition and the properties of the generalized quadratic-phase integral operator and obtain an inversion formula in the class of Boehmians.