Search results for "integral"
showing 10 items of 902 documents
Approximation by step functions of Banach space valued nonabsolute integrals.
2008
The approximation of Banach space valued nonabsolutely integrable functions by step functions is studied. It is proved that a Henstock integrable function can be approximated by a sequence of step functions in the Alexiewicz norm, while a Henstock-Kurzweil-Pettis and a Denjoy-Khintchine-Pettis integrable function can be only scalarly approximate in the Alexiewicz norm by a sequence of step functions. In case of Henstock-Kurzweil-Pettis and Denjoy-Khintchine-Pettis integrals the full approximation can be done if and only if the range of the integral is norm relatively compact. It is also proved that if the target Banach space X does not contain any isomorphic copy of c_0, then the range of t…
The Variational Mcshane Integral in Locally Convex Spaces
2009
The variational McShane integral for functions taking values in a locally convex space is defined, and it is characterized by means of the p-variation of the indefinite Pettis integral
Characterizations of Kurzweil--Henstock--Pettis integrable functions.
2006
We prove that several results of Talagrand proved for the Pettis integral hold true also for the Kurzweil--Henstock--Pettis integral. In particular the Kurzweil--Henstock--Pettis integrability can be characterized by suitable properties of the operators defined by the integrands and by cores of those functions.
Relations among Henstock, McShane and Pettis integrals for multifunctions with compact convex values
2013
Fremlin (Ill J Math 38:471–479, 1994) proved that a Banach space valued function is McShane integrable if and only if it is Henstock and Pettis integrable. In this paper we prove that the result remains valid also in case of multifunctions with compact convex values being subsets of an arbitrary Banach space (see Theorem 3.4). Di Piazza and Musial (Monatsh Math 148:119–126, 2006) proved that if \(X\) is a separable Banach space, then each Henstock integrable multifunction which takes as its values convex compact subsets of \(X\) is a sum of a McShane integrable multifunction and a Henstock integrable function. Here we show that such a decomposition is true also in case of an arbitrary Banac…
A CHARACTERIZATION OF THE WEAK RADON–NIKODÝM PROPERTY BY FINITELY ADDITIVE INTERVAL FUNCTIONS
2009
AbstractA characterization of Banach spaces possessing the weak Radon–Nikodým property is given in terms of finitely additive interval functions. Due to that characterization several Banach space valued set functions that are only finitely additive can be represented as integrals.
Radon-Nikodym derivatives of finitely additive interval measures taking values in a Banach space with basis
2011
Let X be a Banach space with a Schauder basis {en}, and let Φ(I)= ∑n en ∫I fn(t)dt be a finitely additive interval measure on the unit interval [0, 1], where the integrals are taken in the sense of Henstock–Kurzweil. Necessary and sufficient conditions are given for Φ to be the indefinite integral of a Henstock–Kurzweil–Pettis (or Henstock, or variational Henstock) integrable function f:[0, 1] → X.
Variational Henstock integrability of Banach space valued functions
2016
We study the integrability of Banach space valued strongly measurable functions defined on $[0,1]$. In the case of functions $f$ given by $\sum \nolimits _{n=1}^{\infty } x_n\chi _{E_n}$, where $x_n $ are points of a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for Bochner and Pettis integrability of $f$. The function $f$ is Bochner integrable if and only if the series $\sum \nolimits _{n=1}^{\infty }x_n|E_n|$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability of $f$. In this paper we give some conditions for variational Henstock integrability of a…
Measurable selectors and set-valued Pettis integral in non-separable Banach spaces
2009
AbstractKuratowski 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 (n…
Set valued Kurzweil-Henstock-Pettis integral
2005
It is shown that the obvious generalization of the Pettis integral of a multifunction obtained by replacing the Lebesgue integrability of the support functions by the Kurzweil--Henstock integrability, produces an integral which can be described -- in case of multifunctions with (weakly) compact convex values -- in terms of the Pettis set-valued integral.
Differentiation of an additive interval measure with values in a conjugate Banach space
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.