Double integrals and convergence of double series

Abstract

0. I N T R O D U C T I O N The connection between the Perron integral and the convergence of simple series is already well known and its proof can be found in, for example, [MAWl]. A relation between the Lebesgue integral and convergence of double series with positive coefficients has been also discovered. In this paper, we define three new double integrals which allow us to establish a connection between integrability and convergence of arbitrary double series. These integrals a~e based on an integral discovered recently by Buczolich [BU] and Mkhalfi [MKH] and caUed the (GM)-integral. The definition of the (GM)-integral and of other concepts is given in Section 1. Section 2 contains the definition of the new integrals over unbounded intervals and their basic properties, especially a generalisa, tion in two dimensions of Hake's theorem. The last section consists of a proof of the connection mentioned above.