Search results for "Pettis Integral"
showing 10 items of 43 documents
MR2684422 Deville, Robert; Rodríguez, José Integration in Hilbert generated Banach spaces. Israel J. Math. 177 (2010), 285–306. (Reviewer: Luisa Di P…
2010
2010), 285–306, 46Exx (46J10) It is known that each McShane integrable function is also Pettis integrable, while the reverse implication in general is not true. The equivalence of McShane and Pettis integrability depends on the target Banach space X and has been proven: by R. A. Gordon [Illinois J. Math. 34 (1990), no. 3, 557–567, 26A42 (28B15 46G10 49Q15)], and by D. H. Fremlin and J. Mendoza [Illinois J. Math. 38 (1994), no. 1, 127–147, 46G10 (28B05)] if X is separable, by D. Preiss and the reviewer [Illinois J. Math. 47 (2003), no. 4, 1177–1187. 28B05 (26A39 26E25 46G10)] if X=c_0(\Gamma) (for any set \Gamma) or X is super-reflexive, by the second author of the present paper [J. Math. An…
MR2569913: Rodríguez, José. Some examples in vector integration. Bull. Aust. Math. Soc. 80 (2009), no. 3, 384–392. (Reviewer: Luisa Di Piazza),
2009
The paper deals with some classical examples in vector integration due to Phillips, Hagler and Talagrand, revisited from the point of view of the Birkhoff and McShane integrals. More precisely, the author considers: - Phillips' example of a Pettis integrable function f which is not Birkhoff integrable [R. S. Phillips, Trans. Amer. Math. Soc. 47 (1940), 114--145; MR0002707 (2,103c)]. It is proved here that f is universally McShane integrable. - Hagler's example of a scalarly measurable l∞-valued function g which is not strongly measurable. The function g is proved to be universally Birkhoff integrable. - Talagrand's example of a bounded Pettis integrable function φ having no conditional expe…
The McShane, PU and Henstock integrals of Banach valued functions
2002
Some relationships between the vector valued Henstock and McShane integrals are investigated. An integral for vector valued functions, defined by means of partitions of the unity (the PU-integral) is studied. In particular it is shown that a vector valued function is McShane integrable if and only if it is both Pettis and PU-integrable. Convergence theorems for the Henstock variational and the PU integrals are stated. The families of multipliers for the Henstock and the Henstock variational integrals of vector valued functions are characterized.
MR2858094 Musiał, Kazimierz Pettis integrability of multifunctions with values in arbitrary Banach spaces. J. Convex Anal. 18 (2011), no. 3, 769–810.…
2012
Integrals and selections of multifunctions with values in an arbitrary banach space
2017
In this note we will address some recent as well as classical results on multivalued integrals for multifunctions taking values in the hyperspace of convex weakly compact subsets of a general Banach space. In particular the existence of selections integrable in the same sense of the corresponding multifunctions will be considered.
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.