The general theory of integration.
William Henry Young · Proceedings of the Royal Society of London · 1904
Abstract The paper begins with a recapitulation of the well-known definitions of integration and of upper and lower integration (intégral par excès, par défant ; oberes, unteres Integral). The theorem on which the Darboux definition of upper (lower) integration is founded is stated and proved in the following form :— Given any small positive quantity e1, we can determine a positive quantity e, such that, if the fundamental segment S be divided up in any manner into a finite number of intervals, then, provided only the length of each interval is less than e, the upper summation of any function over these intervals differs by less than e1 from a definite limiting value (the upper integral).