Convergence results in Birkhoff weak integrability
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Convergence results in Birkhoff weak integrability Anca Croitoru1
· Alina Gavrilu¸t1
Received: 15 February 2020 / Accepted: 4 April 2020 © Unione Matematica Italiana 2020
Abstract This article presents some properties of real Birkhoff weak integrable functions with respect to a non-additive set function, such as Hölder inequality, Minkowski inequality, Lebesgue type convergence theorems and Fatou type theorem. Keywords Birkhoff weak integral · Hölder inequality · Minkowski inequality · Lebesgue convergence theorem · Fatou theorem · Non-additive set function Mathematics Subject Classification 28A25 · 26A39
1 Introduction Along with Choquet [13], theory of non-additive measures and nonlinear integrals has developed a lot, having important applications in many domains: potential theory, subjective evaluation, optimization, economics, decision making, data mining, artificial intelligence. Regarding non additive measures, a broad presentation of their applications can be found for instance in [40]. The lack of additivity in some aspects of real world has led to the definition of different types of nonlinear integrals (in single-valued or set-valued case). These have been intensively studied both theoretically (e.g., [2,3,5–11,13–27,30–38]) and practically (e.g., [10,12,17,21, 26–29,35,39,40]). The Birkhoff integral [1] was defined for a vector function f : T → X , relative to a complete finite measure m : A → [0, +∞), using series of type ∞ n=0 f (tn )m(Bn ), accordingly to a countable partition {Bn }n∈N of T and tn ∈ Bn , for every n ∈ N. In [15], we introduced and studied a non-linear integral of Birkhoff type (named Birkhoff weak) for vector (real respectively) functions, with respect to a non-additive non-negative
Dedicated to Professor Domenico Candeloro, a sincere friend, collaborator and remarkable Researcher.
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Anca Croitoru [email protected] Alina Gavrilu¸t [email protected]
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Faculty of Mathematics, Alexandru Ioan Cuza University, Carol I Bd. 11, Iasi 700506, Romania
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A. Croitoru, A. Gavrilu¸t
(vector respectively) set function, using finite Riemann type sums and countable partitions. Our definition can be placed between the Birkhoff integral [1] and the Gould integral [22]. In this paper, we expose some properties of the integral introduced in [15]. The structure of the paper is as follows: Sect. 1 is Introduction and Sect. 2 contains some preliminaries. In Sect. 3, we present some classic properties of the Birkhoff weak integral [15] for real functions relative to a non-additive set function. Thus, Hölder inequality, Minkowski inequality, Lebesgue type convergence theorems and Fatou type theorem are established.
2 Preliminaries Let T be a nonempty set, P (T ) the family of all subsets of T and A a σ -algebra of subsets of T . Let N∗ be the set of positive integers. For any A ⊆ T , denote c A = T \A, ℵ A the characteristic function of A and F (T , R) the set of all finite real-valued measurable functions on (T , A). Definition 2.1 Let μ : A → [0, +∞] be a set function with μ(∅) = 0. μ is called:
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