⋄_α-Measurability and combined measure theory on time scales

ABSTRACT

In this paper, we introduce the concept of ${◊}_{\alpha }$-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 $\alpha =1$ and $\alpha =0$, respectively. Moreover, some criteria for ${◊}_{\alpha }$-measurability of a set are obtained and the notion of Lebesgue ${◊}_{\alpha }$-measurable functions is introduced and studied. Based on it, the Lebesgue ${◊}_{\alpha }$-integral and Riemann ${◊}_{\alpha }$-integral are introduced and the relation between them is discussed.

Keywords:

• Measure theory
• ${◊}_{\alpha }$-measurability
• ${◊}_{\alpha }$-measurable functions
• ${◊}_{\alpha }$-integral
• time scales

2010 Mathematics Subject Classifications:

• 26E70
• 28A12
• 28A25

Disclosure statement

The authors declare that they have no competing interests.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant number 11961077].