McShane Integral Research Articles

On coincidence of Pettis and McShane integrability

R.Deville and J.Rodríguez proved that, for every Hilbert generated space X, every Pettis integrable function f: [0, 1] → X is McShane integrable. R.Avilés, G. Plebanek, and J.Rodríguez constructed a weakly compactly generated Banach space X and a scalarly null (hence Pettis integrable) function from [0, 1] into X, which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from [0, 1] (mostly) into C(K) spaces. We focus in more detail on the behavior of several concrete Eberlein (Corson) compact spaces K, that are not uniform Eberlein, with respect to the integrability of some natural scalarly negligible functions from [0, 1] into C(K) in McShane sense.

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.

Descriptive Characterizations of Pettis and Strongly McShane Integrals

Descriptive Characterizations of Pettis and Strongly McShane Integrals

A note of the Mcshane integral on the Riesz space

The main goal of this article is to investigate the integration theory of Mcshane type for functions with values in ordered spaces. Afected by the work of Boccuto, Riecan, and Vrabelova with Kurzweil-Henstock integration we studied the same problems for another important type of integration on such space as Mcshane ones.In this paper we present in other way the definition of Mcshane integral on the Riesz space using a very important lemma of famous Fremlin. In the second section we reconstruct allmost all the propeties of Mchane integral given in [5], [6] and these ones become a little more stronger. We arrive some new results compared with Henstock-Kurzweil ones. In the third section we define the strong version of Mcshane integral and give the neccesary and sufficent condition of this concept. In the fourth section we prove the fundamental theorems of Calculus for the D� -integral and in the fifth we extended an application of this integration to Walsh series .

LEMMA HENSTOCK PADA INTEGRAL M_\alpha

Based on the McShane partition and McShane integral, it can be arranged the partition and integral concepts. Furthermore, it is discussed some simple properties of integral and Lemma Henstock.

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.

ON THE CONVERGENCE THEOREMS OF THE McSHANE INTEGRAL FOR RIESZ-SPACES-VALUED FUNCTIONS DEFINED ON REAL LINE

This paper is a partial result of our researchs in the main topic "On The McShane Integral for Riesz-Spaces-valued Functions Defined on the space ". We have constructed McShane integral for Riesz-spaces-valued functions defined on the space by a technique involving double sequences and proved some basic properties which coincides with the McShane Integral for Banach-spaces valued functions defined on real line. Further, we construct some convergence theorems involving uniformly convergence theorems, monotone convergence theorems and Fatous lemma in the sense of this integral.Keywords : Riesz Space, McShane Integral

A decomposition theorem for the fuzzy Henstock integral

We study the fuzzy Henstock and the fuzzy McShane integrals for fuzzy-number valued functions. The main purpose of this paper is to establish the following decomposition theorem: a fuzzy-number valued function is fuzzy Henstock integrable if and only if it can be represented as a sum of a fuzzy McShane integrable fuzzy-number valued function and of a fuzzy Henstock integrable fuzzy number valued function generated by a Henstock integrable function.

Several comments on the Henstock-Kurzweil and McShane integrals of vector-valued functions

We make some comments on the problem of how the Henstock-Kurzweil integral extends the McShane integral for vector-valued functions from the descriptive point of view.

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.

Integration in Hilbert generated Banach spaces

We prove that McShane and Pettis integrability are equivalent for functions taking values in a subspace of a Hilbert generated Banach space. This generalizes simultaneously all previous results on such equivalence. On the other hand, for any super-reflexive generated Banach space having density character greater than or equal to the continuum, we show that Birkhoff integrability lies strictly between Bochner and McShane integrability. Finally, we give a ZFC example of a scalarly null Banach space-valued function (defined on a Radon probability space) which is not McShane integrable.

Some Alternative Approaches to the McShane Integral

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

SOME EXAMPLES IN VECTOR INTEGRATION

AbstractSome classical examples in vector integration due to Phillips, Hagler and Talagrand are revisited from the point of view of the Birkhoff and McShane integrals.

A negative answer to a problem of Fremlin and Mendoza

This article studies some convergence results for the McShane integral of functions mapping the interval [0, 1] into a Banach space X from the point of view of an open problem proposed by D.H. Fremlin and J. Mendoza in [2], also the anthors give a negative answer to this open problem.

THE STRONG PERRON INTEGRAL IN ℝnREVISITED

It is shown that α-regular McShane and strong Perron integrals considered in [1] are equivalent to the usual McShane integral. This note is related to the recent paper [1], where several Perron and McShane-type integrals in R are introduced and compared. We show here that all those integrals are in fact equivalent to the usual McShane integral. We borrow all the notation and terminology from [1]. The n-dimensional interval I = [a1, b1]× · · · × [an, bn] is said to be α-regular, α ∈ (0, 1), if r(I) = mini(bi − ai) maxi(bi − ai) > α. Let I0 be a fixed interval in R and I the family of all subintervals of I0. With F we denote the free full interval basis F = {(I, x) : I ∈ I, x ∈ I0}. For a given function δ : I0 → (0,∞), called a gauge, and a given α ∈ (0, 1) we define Fδ = { (I, x) ∈ F : I ⊂ U(x, δ(x))}, Fα δ = { (I, x) ∈ F : r(I) > α, I ⊂ U(x, δ(x))}. A finite subset P of Fδ (of Fα δ ) is called an Fδ-division (an Fα δ -division respectively) if for distinct pairs (I1, x1) and (I2, x2) in P, the intervals I1 and I2 are nonoverlapping. If, moreover, ⋃ (I,x)∈P I = I0, then P is called respectively an Fδ-partition and an Fα δ -partition of I0. We recall the definition of the McShane integral. Definition 1. A point function f on I0 is McShane integrable (M-integrable, in brief), with the integral A, if for each 2 > 0 there exists a gauge δ such that ∣∣ ∑ (I,x)∈π f(x)|I| −A ∣∣ < 2 for every Fδ-partition π of I0. Received December 19, 2005. 2000 Mathematics Subject Classification. 26A39.

C-INTEGRAL AND DENJOY-C INTEGRAL

In this paper, we define and study the C-integral of functions mapping an interval [a,b] into a Banach space X and discuss the relations among Henstock integral, C-integral and McShane integral. We also study the Denjoy extension of the C-integral.

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 I 0 ⊂ R m 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 I 0 and satisfies Property (P), then I 0 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 c 0 . We further give an answer to the Karták's question.

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).

THE STRONG PERRON INTEGRAL IN THE n-DIMENSIONAL SPACE ℝn

In this paper, we introduce the SP-integral and the SPfi-integral deflned on an interval in the n-dimensional Euclidean space R n . We also investigate the relationship between these two integrals. It is well known (3) that the Perron integral deflned on an interval of the real line R by major and minor functions which are not assumed to be continuous is equivalent to the one deflned by continuous major and minor functions and that the strong Perron integral deflned on an interval of R by strong major and minor functions is equivalent to the McShane integral. In this paper, we introduce Perron-type integrals deflned on an inter- val of the n-dimensional Euclidean spaceR n using the strong major and minor functions, and investigate the relationship between these integrals. We shall call it the strong Perron integral, or brie∞y SP-integral. For a subset E of the n-dimensional Euclidean spaceR n , the Lebesgue measure of E is denoted by jEj. For a point x = (x1;x2;¢¢¢ ;xn) 2R n , the norm of x is kxk = max1•in jxij and the --neighborhood U(x;-) of x is an open cube centered at x with sides equal to 2-. For an interval I = (a1;b1)£(a2;b2)£¢¢¢(an;bn) ofR n with ai fi(fi 2 (0;1)), then the interval I is said to be fi-regular.

The s-Perron, sap-Perron and ap-McShane Integrals

In this paper, we study the s-Perron, sap-Perron and ap-McShane integrals. In particular, we show that the s-Perron integral is equivalent to the McShane integral and that the sap-Perron integral is equivalent to the ap-McShane integral.