A Note on Convergence Theorems for the Henstock-Kurzweil Integral in Euclidean Space

Abstract

It is known that the controlled convergence theorem is one of the convergence theorems for $\mathcal{HK}$-integrable functions, which deals with nonabsolute $\mathcal{HK}$-integrable functions. This convergence theorem uses the concept of $U AC_\delta^{\star\star}(X)$, 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 $\mathcal{HK}$-integrable functions in the $n$-dimensional space, which may be easier to apply.