Order-type Henstock and McShane integrals in Banach lattice setting

We study Henstock-type integrals for functions defined in a compact metric space $T$ endowed with a regular $\sigma$-additive measure $\mu$, and taking values in a Banach lattice $X$. In particular, the space $[0,1]$ with the usual Lebesgue measure is considered.Comment: 5 page

A note on set-valued Henstock--McShane integral in Banach (lattice) space setting

We study Henstock-type integrals for functions defined in a Radon measure space and taking values in a Banach lattice $X$. Both the single-valued case and the multivalued one are considered (in the last case mainly $cwk(X)$-valued mappings are discussed). The main tool to handle the multivalued case is a R{\aa}dstr\"{o}m-type embedding theorem established in [50]: in this way we reduce the norm-integral to that of a single-valued function taking values in an $M$-space and we easily obtain new proofs for some decomposition results recently stated in [33,36], based on the existence of integrable selections. Also the order-type integral has been studied: for the single-valued case some basic results from [21] have been recalled, enlightning the differences with the norm-type integral, specially in the case of $L$-space-valued functions; as to multivalued mappings, a previous definition ([6]) is restated in an equivalent way, some selection theorems are obtained, a comparison with the Aumann integral is given, and decompositions of the previous type are deduced also in this setting. Finally, some existence results are also obtained, for functions defined in the real interval $[0,1]$.Comment: This work has been modified both as regards the drawing that with regard to the assumptions. A new version is contained in the paper arXiv:1503.0828

Multifunctions determined by integrable functions

Integral properties of multifunctions determined by vector valued functions are presented. Such multifunctions quite often serve as examples and counterexamples. In particular it can be observed that the properties of being integrable in the sense of Bochner, McShane or Birkhoff can be transferred to the generated multifunction while Henstock integrability does not guarantee i

Some convergence theorems for order-Mcshane equi-integral in Riesz space

In this paper we prove some convergence theorems of order-Macshane equi -integrals on Banach lattice and arrive same result in L-space as on Mcshane norm-integrals

Decompositions of Weakly Compact Valued Integrable Multifunctions

We give a short overview on the decomposition property for integrable multifunctions, i.e., when an "integrable in a certain sense" multifunction can be represented as a sum of one of its integrable selections and a multifunction integrable in a narrower sense. The decomposition theorems are important tools of the theory of multivalued integration since they allow us to see an integrable multifunction as a translation of a multifunction with better properties. Consequently, they provide better characterization of integrable multifunctions under consideration. There is a large literature on it starting from the seminal paper of the authors in 2006, where the property was proved for Henstock integrable multifunctions taking compact convex values in a separable Banach space X. In this paper, we summarize the earlier results, we prove further results and present tables which show the state of art in this topi