Convergence Theorems in Interval-Valued Riemann–Lebesgue Integrability

Abstract

We provide some limit theorems for sequences of Riemann–Lebesgue integrable functions. More precisely, Lebesgue-type convergence and Fatou theorems are established. Then, these results are extended to the case of Riemann–Lebesgue integrable interval-valued multifunctions.

Inequalities in Riemann–Lebesgue Integrability

“…In this section, we present some inequalities regarding the Riemann-Lebesgue integral. In the following, S is a set with cardS ≥ ℵ 0 , (X, ∥ • ∥) is a Banach space and m : C → [0, ∞) is a non-additive measure in the class M. For more results on this subject, we refer to [13].…”

“…Finally, ref. [24] (Remark 2) contains a summary of known results related to the properties of the space X.…”

“…According to [24] (Remark 4), for p ∈ [1, ∞), the function ∥ • ∥ p is a seminorm on the linear space of measurable RL m -integrable functions.…”

“…In recent decades, some generalizations of different classic inequalities have been obtained, such as Cauchy, Hölder, Minkowski, Jensen, Hardy, Steffensen, Jensen-Steffensen, Cebyshev, Clarkson, Riemann-Liouville, etc. (for instance, [4,[7][8][9][10][11][12][13][14]). Furthermore, some integral inequalities have been established for multifunctions in different integral settings (for example, [15][16][17][18]).…”

Convergence Theorems for Varying Measures Under Convexity Conditions and Applications

“…Convergence results for varying measures have significant applications to various fields of pure and applied sciences including stochastic processes, statistics, control and game theories, transportation problems, neural networks, signal and image processing (see, for example, [2,[5][6][7][8]15,20,24,28,34]). E.T.…”