McShane Integral Research Articles

Vector-valued McShane and strong McShane integrals with the Alexiewicz norm and its Riesz representation theorems

Vector-valued McShane and strong McShane integrals with the Alexiewicz norm and its Riesz representation theorems

Radon–Nikodým theorems in non-separable Banach spaces

In this paper, we present two Radon–Nikodým theorems in non-separable Banach spaces using Pettis and variational McShane integrals. The first one works for a dominated additive interval multifunction \Phi : \mathcal{I}\to ck(X) , and the second one works for a dominated strong multimeasure M : \mathcal{L}\to cwk(X) , where \mathcal{I} is the family of all closed non-degenerate subintervals of the interval W = [0,1]^{m}\subset \mathbb{R}^{m} , \mathcal{L} is the family of all Lebesgue measurable subsets of W , and cwk(X) (ck(X)) is the family of all convex weakly compact (convex compact) non-empty subsets of X .

La integral, una visión de su evolución a través del tiempo III

In this opportunity we present some areas of the evolution of the integral, which complement what was deal with in parts I and II. Thus we give a conceptual overview of the McShane integral, of the C-integral, of the Bochner integral, of the Lm HK integral , of the integral on manifolds and on a modern point of view of the Riemann integral.

Multipliers of Variationally Mcshane Integrable Functions in Locally Convex Space

Abstract We study measurable real valued multipliers of variationally McShane (resp. McShane) integrable functions defined on a σ-finite outer regular quasi-Radon measure space and taking values in a complete locally convex topological vector space X. We also show: in case X is representable by semi-norm then essentially bounded real measurable functions are multipliers of functions which are Pettis integrable as well as integrable by semi-norm. The space of real valued measurable and essentially bounded functions turn out to be precisely the multipliers of variationally McShane (resp. McShane) integrable functions in case X is representable by semi-norm.

Vector Valued Multipliers of McShane Integrable Functions

We study the algebra of vector valued multipliers of Banach algebra valued McShane integrable functions. We prove that if X is a commutative Banach algebra, with identity e of norm one, then functions associated with measures of strong bounded variation and the set {L?([a, b],?) e} are vector valued multipliers of McShane integrable functions. We find some necessary and another set of sufficient conditions for a functiong to define a multiplier. In case X satisfies Radon Nikodym property (weak Radon Nikodym property), we study multiplier operators.

A New fuzzy McShane integrability

Abstract We introduce the notion of the fuzzy McShane integral in the linear topology sense and we discuse its relation with the fuzzy Pettis integral introduced recently by Chun-Kee Park in [On the Pettis integral of fuzzy mappings in Banach spaces, Commun. Korean Math. Soc. 22 (2007), 535–545].

On the Differentiation of Henstock and McShane Integrals

It is well known that the derivative of the primitive of 1-dimensional Henstock integral exists almost everywhere. Point-interval pairs used in the derivative are Henstock point-interval pairs, which are consistent with point-interval pairs used in the Henstock integral. Note that “almost everywhere” is a set of points, more precisely, the derivative does not exist on a set of points with measure zero. We can transform a set of Henstock point-interval pairs to a set of points with measure zero because of Vitali’s covering theorem. For 1-dimensional McShane integrals, [Formula: see text]-dimensional McShane and Henstock integrals, covering theorems of Vitali’s type cannot be applied. In this paper, we shall discuss differentiation of [Formula: see text]-dimensional McShane and Henstock integrals.

Dunford–Henstock–Kurzweil and Dunford–McShane Integrals of Vector-Valued Functions Defined on m-Dimensional Bounded Sets

In this paper, we define the Dunford–Henstock–Kurzweil and the Dunford–McShane integrals of Banach space-valued functions defined on a bounded Lebesgue measurable subset of m-dimensional Euclidean space $${\mathbb {R}}^{m}$$ . We will show that the new integrals are “natural” extensions of the McShane and the Henstock–Kurzweil integrals from m-dimensional closed non-degenerate intervals to m-dimensional bounded Lebesgue measurable sets. As applications, we will present full descriptive characterizations of the McShane and Henstock–Kurzweil integrals in terms of our integrals. Moreover, a relationship between new integrals will be proved in terms of the Dunford integral.

McShane Integrability Using Variational Measure

If f : [a, b] → R is McShane integrable on [a, b], then f is McShane integrable on every Lebesgue measurable subset of [a, b]. However, integrability of a real-valued function on [a, b] does not imply McShane integrability on any E ⊆ [a, b]. In this paper, we give a characterization for the McShane integrability of f : [a, b] → R over E ⊆ [a, b] using concept of variational measure.

Double Lusin condition and Vitali convergence theorem for the Itô–McShane Integral

In this paper, we formulate a version of Vitali convergence theorem for the Ito–McShane integral of an operator-valued stochastic process with respect to a Q-Wiener process. We also characterize the integral using double Lusin condition.

Strongly measurable functions and multipliers of $${\mathcal {M}}-$$ integrable functions

We investigate the integrability of Banach space valued strongly measurable functions defined on [0, 1]. In the case of functions given by $$f=\sum _{n=1}^{\infty }x_n \chi _{E_n}$$ where $$x_n\in X$$ and the sets $$E_n$$ are Lebesgue measurable and pairwise disjoint subsets of [0, 1], there are well known characterizations for Bochner and McShane integrability of f. The absolute (resp. unconditional) convergence of $$\sum _{n=1}^{\infty }x_n m(E_n)$$ is equivalent to Bochner (resp. McShane) integrability of f. We give some conditions for scalar McShane and weakly McShane integrability of f and their relation with unconditionally converging operators. We also study the space of vector valued multipliers of strongly McShane $$(\mathcal {SM})-$$ integrable functions. We prove that if X is a commutative Banach algebra, with identity e of norm one, satisfying Radon–Nikodym property and $$M: \mathcal {SM}\rightarrow \mathcal {SM}$$ is a bounded linear operator, then there exists $$g\in L^\infty ([0,1],X)$$ such that $$M(f)= fg$$ for all $$f\in \mathcal {SM}$$ . Some results on multipliers of McShane $$({\mathcal {M}})-$$ integrable functions are also derived.

The Weak Integral by Partitions of Unity

We introduce the notion of the the weak integral by partitions of unity for functions defined on a \(\sigma\)-finite outer regular quasi Radon measure space \((S,\Sigma,\mathcal{T},\mu)\) into a Banach space \(X\) and discuss its relation with the weak McShane integral which has been introduced by M. Saadoune and R. Sayyad (2014).

English

We present the full descriptive characterizations of the strong McShane integral (or the variational McShane integral) of a Banach space valued function $f\colon W \to X$ defined on a non-degenerate closed subinterval $W$ of $\mathbb {R}^{m}$ in terms of strong absolute continuity or, equivalently, in terms of McShane variational measure $V_{\mathcal {M}} F$ generated by the primitive $F\colon \mathcal {I}_{W} \to X$ of $f$, where $\mathcal {I}_{W}$ is the family of all closed non-degenerate subintervals of $W$.

English

We characterize the weak McShane integrability of a vector-valued function on a finite Radon measure space by means of only finite McShane partitions. We also obtain a similar characterization for the Fremlin generalized McShane integral.

English

english

On McShane-Stieltjes Integrals of Interval-valued Functions and Fuzzy-number-valued Functions on Time Scales

In this paper, we introduce the notion of the McShane-Stieltjes (MS) integrals of interval-valued functions and fuzzy-number-valued functions on time scales which are extensions of the McShane (M) integrals of interval-valued functions and fuzzy-number-valued functions on time scales [3] and investigate some of theirproperties.

English

The multiplier for the weak McShane integral which has been introduced by M. Saadoune and R. Sayyad (2014) is characterized.

The New Extensions of the Henstock–Kurzweil and the McShane Integrals of Vector-Valued Functions

In this paper, we define the Hake–Henstock–Kurzweil and the Hake–McShane integrals of Banach space valued functions defined on an open and bounded subset G of m-dimensional Euclidean space $$\mathbb {R}^{m}$$ . These are ”natural” extensions of the McShane and the Henstock–Kurzweil integrals from m-dimensional closed non-degenerate intervals to bounded and open subsets of $$\mathbb {R}^{m}$$ . Our goal is not a generalization for the sake of generalization. Indeed, we will show theorems which reduce the study of our integrals to the study of McShane and Henstock–Kurzweil integrals. As applications, we will present Hake-type theorems for the Henstock–Kurzweil and the McShane integrals in terms of our integrals.

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 Lp 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 L1 spaces) and Lebesgue-Bochner spaces (topological and structural properties, scalar convergence).

Properties of the McShane Integrability

In 1990 Gordon[12] introduced the concepts of the McShane integral of Banach-valued functions. This integral is a generalized Riemann integral of functions which have values in a Banach space. For real-valued functions the McShane integral and the Lebesgue integral are equivalent. Gordon[12] and Fremlin and Mendoza[3] have developed the properties of this integral. In this paper, we will investigate some properties and relations among the McShane integral, the Dunford integral, and the Pettis integral. Let a given separable  contain no copy of C 0 and T be a weakly compact operator. Then we will show that a Dunford integrable function f:[a,b] →X is intrinsically-separable valued when f is McShane integrable. Also, we will show that if there exists a sequence (f n ) of McShane integrable functions from [a,b] to X such that for each x ? ∈X ? ,x ? f n → x ? f a·e·, then f is McShane integrable. Finally, we will show that if f :[a,b] →X is McShane integrable, then {x ? f x ? ∈Bx ? } is uniformly integrable in L¹(μ) where X contains no copy of C 0 .