A NOTE ON CONVERGENCE THEOREMS FOR THE HENSTOCK-KURZWEIL INTEGRAL IN EUCLIDEAN SPACE

It is known that the controlled convergence theorem is one of the convergence theorems for HK-integrable functions, which deals with nonabsolute HK-integrable functions. This convergence theorem uses the concept of UAC(delta)(star star) (X), and the proof of the convergence theorem in the n-dimensional space is rather involved. In this paper, we shall give a necessary and sufficient condition for UAC(delta)(star star) (X), which is in terms of Lebesgue integrable functions on X and primitive functions, see Theorem 9. Furthermore, we shall give three convergence theorems, Theorems 5, 7 and 10, for HK-integrable functions in the n-dimensional space, which may be easier to apply.