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Real analysis. An introduction to the theory of real functions and integration. (English) Zbl 0978.28001
Studies in Advanced Mathematics. Boca Raton, FL: Chapman & Hall/CRC. xiv, 567 p. £67.00; $ 99.99 (2001).

The subjects of this textbook on abstract analysis are general topology, the theory of measure and integration, and some particular topics related to measure theory. Correspondingly, the book consists of three parts. The first part begins with a chapter on set theory, including discussions of the axiom of choice, cardinality, etc. The second chapter deals with metric spaces and precedes, didactically well organized, the following chapter on point-set topology.

A chapter on measurable spaces and measurable functions opens the second part of the book. The next topic are measures where a particularly broad discussion of Carathéodory’s extension procedure is presented. Clearly, the following chapter is devoted to integration, encompassing in particular a discussion of the relation between the Riemann and Lebesgue integral as well as the theorems of Radon-Nikodým and Fubini. In a further chapter, the author gives a very brief treatment of differentiation in Banach spaces and an appropriate presentation of the change of variables formula for the Lebesgue integral in 𝐑 n .

Among the topics the third part deals with are signed and complex measures, L p spaces, as well as Radon measures on locally compact Hausdorff spaces and the Riesz’s representation theorem. The final chapter of the book concerns the classical theory of real one-variable functions, as indicated by the key words monotone functions, functions of bounded variation, absolute continuity and singularity.

The author presents his subjects very thoroughly and in most cases in great detail, the presentation consequently being less concise. The book is suitable as a textbook and as a reference source, as well; it supplements other well-known classical textbooks on real analysis. The reader finds many interesting historical remarks and, at the end of each section, various exercise problems and a summary of the new terms introduced. However, the bibliography containing eight references only is completely poor and inappropriate for such a book (even though this may be a little bit compensated by the above mentioned enlightening historical references and comments).

28-01Textbooks (measure and integration)
26-01Textbooks (real functions)