⋄_α-Measurability and combined measure theory on time scales

In this paper, we introduce the concept of (Formula presented.) -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 (Formula presented.) and (Formula presented.), respectively. Moreover, some criteria for (Formula presented.) -measurability of a set are obtained and the notion of Lebesgue (Formula presented.) -measurable functions is introduced and studied. Based on it, the Lebesgue (Formula presented.) -integral and Riemann (Formula presented.) -integral are introduced and the relation between them is discussed.