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.
In order to give an adequate account of the subject in its present condition, a large amount of new matter has had to be introduced; and this has made it necessary to divide the treatise into two volumes. The matter contained in the first edition has been carefully revised, amplified, and in many cases rewritten.
The parts of the subject which were dealt with in the first five chapters of the first edition have been expanded into the eight chapters of the present first volume of the new edition. With a view to greater unity of treatment of the Theory of Integration, some theorems which appeared in Chapter VI, of the first edition, have however been included in the present volume. A considerable part of the Theory of Integration, in relation to series and sequences, still however remains for treatment in Volume II.
On controversial matters connected with the fundamentals of the Theory of Aggregates, the considerable diversity of opinion which has arisen amongst Mathematicians has been taken into account, but in general no attempt has been made to give dogmatic decisions between opposed opinions. In view of the delicate questions which arise as to the legitimacy and meaning of the axiom known as the Multiplicative Axiom, or as the Principle of Zermelo, the policy has been adopted of so framing the proofs of theorems as to avoid an appeal to the axiom, whenever that course appeared to be possible; in other cases, the necessity for the employment of the axiom has been pointed out.
Ample references to sources of information are given throughout, but such references do not provide the means for compiling a complete list of writings on the subject. No attempt has been made to settle questions of priority of discovery.
My thanks are due to Dr H. F. Baker, F.R.S., Lowndean Professor of Astronomy and Geometry, in the University of Cambridge, who has kindly read nearly all the proofs as they passed through the Press.