Properties of the Riemann–Lebesgue integrability in the non-additive case

The Riemann-Lebesgue Integral of Interval-Valued Multifunctions

“…(vi) exhaustive if lim n→∞ m(A n ) = 0, for every sequence of pairwise disjoint sets (A n ) n∈N ⊂ A. Moreover m satisfies property (σ) if the ideal of m-zero sets is stable under countable unions (see for example [34], Definition 2.3).…”

“…We recall the following definition for the integrable Banach-valued functions f : S → X with respect to non-negative measures given in [38,39]: Definition 4. A function f is called unconditional Riemann-Lebesgue (RL ) m-integrable (on S) if there exists b ∈ X such that for every ε > 0, there exists a countable partition P ε of S, so that for every countable partition P = {A n } n∈N of S with P ≥ P ε , f is bounded on every A n , with m(A n ) > 0 and for every t n ∈ A n , n ∈ N, the series ∑ +∞ n=0 f (t n )m(A n ) is unconditional convergent and For the properties of this integral with respect to a submeasure we refer to the results given in [34]. Moreover we have that Proposition 1.…”

“…In the literature several methods of integration for functions and multifunctions have been studied extending the Riemann and Lebesgue integrals. In this framework a generalization of Riemann sums was given in [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] while another generalization is due to Kadets and Tseytlin [38], who introduced the absolute Riemann-Lebesgue |RL| and unconditional Riemann-Lebesgue RL integrability, for Banach valued functions with respect to countably additive measures. They proved that in finite measure space, the Bochner integrability implies |RL| integrability which is stronger than RL integrability that implies Pettis integrability.…”

“…They proved that in finite measure space, the Bochner integrability implies |RL| integrability which is stronger than RL integrability that implies Pettis integrability. Regarding this last extension contributions are given also in [21,23,34,39].…”

“…In this paper, motivated by the large number of fields in which the interval-valued multifunction can be applied, we introduce a new type of integral of an interval-valued multifunction G with respect to an interval-valued submeasure M with respect to the weak interval order relation introduced in [4] by Guo and Zhang. Although the construction procedure of the integral is similar to the one given in [34,38,39], the integral proposed is a generalization of it since we are concerned with the study of a Riemann-Lebesgue set-valued integrand with respect to an arbitrary interval-valued set function, not necessarily countably additive. So the novelty of this construction concerns not only the codomain of the integrands but also the non-additivity of the measure with respect to which they are integrated.…”