On the new theory of integration

Abstract

1. In a paper published in the ' Proceedings of the London Mathematical Society,* addressed to persons already acquainted with Lebesgue integration, I endeavoured to show that the method of monotone sequences enabled us to recognise intuitively the extensibility to Lebesgue integration of results known to be true for Riemann integrals. For this purpose I naturally employed known results in the proofs of Sets of joints ; and, of course, also pre-supposed the proofs of the classical theorems whose generalisation was in question. In the present communication I propose to indicate briefly how the method of monotone sequences enables us to prove, at one and the same time, these theorems and their generalisations. For this purpose we have only to employ a slight modification of the procedure indicated in the paper cited ; one which, however, avoids all reference to the Theory of Sets of Points, and assumes no results whatever in the Theory of Integration.