关键词: δ-fine division, KH-integral, LSRS, ULSRS

Abstract: The well-known Controlled Convergence Theorem^[5] and the equi-integrabillty theorem^[9] are the main convergence theorems of the Kurzweil-Henstock integral,which is of the non-absolute type.These theorems are fundamential in the application of the KH-integral to real and functional analysis.But their conditions can be weakened to extend their applications.
In this paper, using the property of Locally-Small-Riemann-Sums^[7], we give all other convergence theorem (Theorem 1).By Theorem 2 we prove that Theorem 1 contains the Equi-integrability Theorem and is not equivalent to it. Therefore the Controlled Convergence Theorem and the Equi-integrabillty Theorem are all corollaries of Theorem 1.

Key words: δ-fine division, KH-integral, LSRS, ULSRS