General theory of functions and integration / Angus E. Taylor

This is a valuable addition to the growing collection of textbooks, addressed to students at the upper-undergraduate-to-graduate level, in what has generally come to be called "real analysis''.
The author feels that too often a student preparing himself for advanced graduate study and research in analysis finds himself at a disadvantage because his study of integration has been largely from a single point of view. A primary aim of the present book is to clarify for the student the various approaches to modern integration theory and their interrelations, especially the classical Lebesgue approach, the Daniell approach, and the general setting for integration appropriate in functional analysis.
The chapter titles give some indication of the subjects treated and the arrangement: The real numbers—point sets and sequences; Euclidean space—topology and continuous functions; Abstract spaces; The theory of measure; The Lebesgue integral; Integration by the Daniell method; Iterated integrals and Fubini's theorem; The theory of signed measures; Functions of one real variable. Most of the "name'' theorems one would expect to find are included: the Stone-Weierstrass theorem, the Radon-Nikodým theorem, etc. Zorn's lemma and a few other forms of the maximality principle are discussed briefly and used occasionally.
It seems to the reviewer that the author has succeeded admirably in his attempt to produce "a judicious blend of the particular and the general, of the concrete and the abstract, as an aid to graduate students and as a guide to the further expansion of their mathematical horizons''. It will be viewed as a defect by some that the treatment of convergence is almost entirely restricted to that of sequences. Scattered throughout are brief but informative historical remarks and references to original sources. These form a pleasant feature of the author's generally attractive and lucid style. The numerous problems and challenges to the student to supply details of arguments in the text appear to have been selected with fine judgment. (MathSciNet)

Bibliogr. p. 423-427. Index