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The first three editions of H.L.Royden's Real Analysis have contributed to the education of
generations of mathematical analysis students. This fourth edition of Real Analysis preserves
the goal and general structure of its venerable predecessors-to present the measure theory,
integration theory, and functional analysis that a modem analyst needs to know.

The book is divided the three parts: Part I treats Lebesgue measure and Lebesgue
integration for functions of a single real variable; Part II treats abstract spaces-topological
spaces, metric spaces, Banach spaces, and Hilbert spaces; Part III treats integration over
general measure spaces, together with the enrichments possessed by the general theory in
the presence of topological, algebraic, or dynamical structure.
The material in Parts II and III does not formally depend on Part I. However, a careful
treatment of Part I provides the student with the opportunity to encounter new concepts in a
familiar setting, which provides a foundation and motivation for the more abstract concepts
developed in the second and third parts. Moreover, the Banach spaces created in Part I, the
LP spaces, are one of the most important classes of Banach spaces. The principal reason for
establishing the completeness of the LP spaces and the characterization of their dual spaces
is to be able to apply the standard tools of functional analysis in the study of functionals and
operators on these spaces. The creation of these tools is the goal of Part II.

This edition contains 50% more exercises than the previous edition
Fundamental results, including Egoroff s Theorem and Urysohn's Lemma are now
proven in the text.
The Borel-Cantelli Lemma, Chebychev's Inequality, rapidly Cauchy sequences, and
the continuity properties possessed both by measure and the integral are now formally
presented in the text along with several other concepts.
There are several changes to each part of the book that are also noteworthy:
Part I

The concept of uniform integrability and the Vitali Convergence Theorem are now

presented and make the centerpiece of the proof of the fundamental theorem of
integral calculus for the Lebesgue integral
A precise analysis of the properties of rapidly Cauchy sequences in the LP(E) spaces,
1 < p < oo, is now the basis of the proof of the completeness of these spaces
Weak sequential compactness in the LP(E) spaces, 1 < p < oo, is now examined in
detail and used to prove the existence of mi...
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