The time which has elapsed since the publication of the first edition of this treatise has been a period of great activity in the development of the
Theory of Functions of a real variable. In particular, the introduction of the Lebesgue Integral, which was new in 1907, has since produced its full effect, in the generalization of the Theory of Integration, and upon the theory of the representation of functions by means of Fourier's, and other, series and integrals.