⋄α-Measurability and combined measure theory on time scales. Issue 8 (24th May 2022) - Wang, Chao; Qin, Guangzhou; Agarwal, Ravi P.; O'Regan, Donal
ABSTRACT: In this paper, we introduce the concept of ◊ α -measurability and establish the combined measure theory on time scales. Particularly, one can obtain Δ-measure theory and ∇-measure theory on time scales by letting α = 1 and α = 0, respectively. Moreover, some criteria for ◊ α -measurability of a set are obtained and the notion of Lebesgue ◊ α -measurable functions is introduced and studied. Based on it, the Lebesgue ◊ α -integral and Riemann ◊ α -integral are introduced and the relation between them is discussed.