171,478 research outputs found
A remark on weak McShane integral
No full textsummary: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
On coincidence of Pettis and McShane integrability
No full textsummary:R. Deville and J. Rodríguez proved that, for every Hilbert generated space , every Pettis integrable function is McShane integrable. R. Avilés, G. Plebanek, and J. Rodríguez constructed a weakly compactly generated Banach space and a scalarly null (hence Pettis integrable) function from into , which was not McShane integrable. We study here the mechanism behind the McShane integrability of scalarly negligible functions from (mostly) into spaces. We focus in more detail on the behavior of several concrete Eberlein (Corson) compact spaces , that are not uniform Eberlein, with respect to the integrability of some natural scalarly negligible functions from into in McShane sense
The Vitali convergence theorem for the vector-valued McShane integral
No full textsummary:The classical Vitali convergence theorem gives necessary and sufficient conditions for norm convergence in the space of Lebesgue integrable functions. Although there are versions of the Vitali convergence theorem for the vector valued McShane and Pettis integrals given by Fremlin and Mendoza, these results do not involve norm convergence in the respective spaces. There is a version of the Vitali convergence theorem for scalar valued functions defined on compact intervals in given by Kurzweil and Schwabik, but again this version does not consider norm convergence in the space of integrable functions. In this paper we give a version of the Vitali convergence theorem for norm convergence in the space of vector-valued McShane integrable functions
New extension of the variational McShane integral of vector-valued functions
No full textsummary:We define the Hake-variational McShane integral of Banach space valued functions defined on an open and bounded subset of -dimensional Euclidean space . It is a "natural" extension of the variational McShane integral (the strong McShane integral) from -dimensional closed non-degenerate intervals to open and bounded subsets of . We will show a theorem that characterizes the Hake-variational McShane integral in terms of the variational McShane integral. This theorem reduces the study of our integral to the study of the variational McShane integral. As an application, a full descriptive characterization of the Hake-variational McShane integral is presented in terms of the cubic derivative
The s-Perron, sap-Perron and ap-McShane integrals
No full textsummary: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
Some remarks on descriptive characterizations of the strong McShane integral
No full textsummary:We present the full descriptive characterizations of the strong McShane integral (or the variational McShane integral) of a Banach space valued function defined on a non-degenerate closed subinterval of in terms of strong absolute continuity or, equivalently, in terms of McShane variational measure generated by the primitive of , where is the family of all closed non-degenerate subintervals of
On the strong McShane integral of functions with values in a Banach space
No full textsummary:The classical Bochner integral is compared with the McShane concept of integration based on Riemann type integral sums. It turns out that the Bochner integrable functions form a proper subclass of the set of functions which are McShane integrable provided the Banach space to which the values of functions belong is infinite-dimensional. The Bochner integrable functions are characterized by using gauge techniques. The situation is different in the case of finite-dimensional valued vector functions
Super McShane identity
No full textThe authors derive a McShane identity for once-punctured super tori. Relying upon earlier work on super Teichm\"uller theory by the last two-named authors, they further develop the supergeometry of these surfaces and establish asymptotic growth rate of their length spectra
Some full characterizations of the strong McShane integral
No full textsummary:Some full characterizations of the strong McShane integral are obtained
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