On generalized convergence theorems for Thomson’s ℬ-integral on ℝm

Abstract

In this paper, we extend the \({\mathcal{B}}\)-integral on \(\mathbb{R}\) to one on \(\mathbb{R}^{m}\). After proving a weaker version of the equiintegrability theorem, the equivalence of five generalized convergence theorems will be established, which include the equiintegrability theorem, two versions of the generalized controlled convergence theorem which are based on \(UACG^\nabla\) and \(UACG_{\mathcal{B}}^{\ast\ast}\), respectively, the generalized variational convergence theorem and the uniformly \(MGSRS_{\mathcal{B}}\) (modified \(GSRS\) with respect to \({\mathcal{B}}\)) convergence theorem.