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In this paper convergence theorems for sequences of scalar, vector and multivalued Pettis integrable functions on a topological measure space are proved for varying measures vaguely convergent., 21 pages

The aim of this paper is to investigate different types of multi-integrals of finite variation and to obtain decomposition results., 9 pages

We study the space of vector valued multipliers of strongly Henstock-Kurzweil \((\mathcal{SHK})\) integrable functions. We prove that if \(X\) is a commutative Banach algebra, with identity \(e\) of norm one, satisfying Radon-Nikodym property and \(g:[a,b] \rightarrow X\) is of strong bounded variation, then the multiplication operator defined by \(M_g(f)=fg\) maps \(\mathcal{SHK}\) to \(\mathcal{SHK}.\) We also investigate the problems when the domain is \(\mathcal{HK}\) or when \(X\) satisfies weak Radon-Nikodym property.

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).

We present a complete characterization of finitely additive interval measures with values in conjugate Banach spaces which can be represented as Henstock-Kurzweil-Gelfand integrals. If the range space has the weak Radon-Nikodým property (WRNP), then we precisely describe when these integrals are in fact Henstock-Kurzweil-Pettis integrals.

A new concept of McShane-tightness is introduced to pass from scalar (alias weak) McShane integrability to Pettis integrability. It is used also to derive a scalar McShane version of the Vitali theorem.

In this paper we study the application of a general integral called a distributional Henstock-Kurzweil integral which includes the Henstock-Kurzweil integral and the wide Denjoy integral. We prove the existence of solutions of the wave equation involving the distributional Henstock-Kurzweil integral by using Schauder's fixed-point theorem.

In this paper we discuss the relationship between the Kurzweil-Henstock-Pettis and Kurzweil-Henstock integrals in Banach spaces. We prove that in Schur spaces the Kurzweil-Henstock-Pettis and Kurzweil-Henstock integrability for measurable functions satisfying the condition $(C)$ are equivalent. In particular, in Schur spaces the Kurzweil-Henstock-Dunford, Kurzweil-Henstock-Pettis and Kurzweil-Henstock integrability for measurable functions satisfying the condition $(C)$ are equivalent.

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

A characterization of Banach spaces possessing the Radon–Nikodým property is given in terms of finitely additive interval functions. We prove that a Banach space $X$ has the RNP if and only if each $X$-valued finitely additive interval function possessing absolutely continuous variational measure is a variational Henstock integral of an $X$-valued function. Due to that characterization several $X$-valued set functions that are only finitely additive can be represented as integrals.

In this paper we discuss integration by parts for several generalizations of the Riemann-Stieltjes integral. In addition, we obtain new results on integration by parts for the Henstock-Stieltjes integral and its interior modification for Banach space-valued functions.

In this paper properties of Kurzweil-Henstock and Kurzweil-Henstock-Pettis integrals for vector-valued functions are studied. In particular, the absolute integrability for Kurzweil-Henstock integrable functions is characterized and a Kurzweil-Henstock version of the Vitali Theorem for Pettis integrable functions is given.

We give a partial answer to the question in the title by showing that the McShane and Pettis integrals coincide for functions with values in super-reflexive spaces as well as for functions with values in $c_0(\Gamma)$. We also improve an example of Fremlin and Mendoza, according to which these integrals do not coincide in general, by showing that, at least under the Continuum Hypothesis, there is a scalarly negligible function which is not McShane integrable.

In this paper, we define and study the McShane integral of functions mapping a compact interval $I_0$ in $R^m$ into a Banach space $X$. We compare this integral with the Pettis integral and prove, in particular, that the two integrals are equivalent if $X$ is reflexive and the unit ball of the dual $X^*$ satisfies an additional condition (P). This gives additional information on an implicitly stated open problem of R.A. Gordon and on the work of D.H. Fremlin and J. Mendoza.

This note concerns the generalized Riemann integrals using free tagged subdivisions, and leading to absolute integration theories. We first establish an approximation theorem by step functions. This result will allow us to define a natural notion of equi-integrability in Lebesgue space $L^1$. Some applications, as a new caracterization of compact parts of $L^1$, are presented.

We shall develop the properties of an integral for Banach-valued functions. The formalism is the generalized Riemann integral introduced by Kurzweil \cite{Kur} and Henstock \cite{Hen}. More precisely, the presentation is close to the McShane approach \cite{McS}. Besides its simplicity of presentation, four advantages characterize this theory: %{\leftskip=1cm \item{(i)} the definition can be used for real-valued functions, and can be generalized without modification to general real and complex Banach spaces; \item{(ii)} when a function is integrable its norm is also integrable, and the proof is straightforward from the definition; \item{(iii)} for finite dimension spaces the theory is equivalent to the McShane's theory, which is itself equivalent to the Lebesgue's theory; \item{(iv)} and lastly, for general Banach space, we can prove the equivalence to the Bochner's theory. \par%}

In the present paper, some Olech and Visintin-type results are obtained in Henstock-Kurzweil-Pettis integration. More precisely, under extreme or denting point condition, one can pass from weak convergence (i.e. convergence with respect to the topology induced by the tensor product of the space of real functions of bounded variation and the topological dual of the initial Banach space) or from the convergence of integrals to strong convergence (i.e. in the topology of Alexiewicz norm or, even more, of Pettis norm). Our results extend the results already known in the Bochner and Pettis integrability setting.

The McShane integral of Banach space-valued functions $f:I\to X$ defined on an $m$-dimensional interval $I$ is considered in this paper. We show that a McShane integrable function is integrable over measurable sets contained in $I$ (Theorem 9). A certain type of absolute continuity of the indefinite McShane integral with respect to Lebesgue measure is derived (Theorem 11) and we show that the indefinite McShane integral is countably additive (Theorem 16). Allowing more general partitions using measurable sets instead of intervals another McShane type integral is defined and we show that it is equivalent to the original McShane integral (Theorem 21)

The main goal of the paper is to define new type of integrals for vector-valued functions on time scales. This allows to make possible the advantages of dynamic equations also for vector-valued functions i.e. for dynamic modeling in Banach spaces. To do it we define some appropriate integrals for vector-valued functions on time scales and we prove their properties. We emphasize on the particular ones, which are useful for solving dynamic equations in Banach spaces.