The well-known Controlled Convergence Theorem([5]) and the equi-integrability theorem([9]) are the main convergence theorems of the Kurzweil-Henstock integral, which is of the non-absolute type. These theorems are fun...The well-known Controlled Convergence Theorem([5]) and the equi-integrability theorem([9]) are the main convergence theorems of the Kurzweil-Henstock integral, which is of the non-absolute type. These theorems are fundamental 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 an 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-integrability Theorem are all corollaries of Theorem 1.展开更多
文摘The well-known Controlled Convergence Theorem([5]) and the equi-integrability theorem([9]) are the main convergence theorems of the Kurzweil-Henstock integral, which is of the non-absolute type. These theorems are fundamental 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 an 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-integrability Theorem are all corollaries of Theorem 1.
