Accomplishments[ edit ] In a series of papers reprinted in his Algebraic Logic, Halmos devised polyadic algebras , an algebraic version of first-order logic differing from the better known cylindric algebras of Alfred Tarski and his students. An elementary version of polyadic algebra is described in monadic Boolean algebra. In addition to his original contributions to mathematics, Halmos was an unusually clear and engaging expositor of university mathematics. He won the Lester R. Ford Award in [5] and again in shared with W.

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Halmos C. Moore Paul R. Halmas C. Graduate texts in mathematics, 18 Reprint of the ed. Bibliography: p. Measure theory. Se ries. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag.

Softcover reprint ofthe hardcover 1st edition PREFACE My main purpose in this book is to present a unified treatment of that part of measure theory which in recent years has shown itself to be most useful for its applications in modern analysis. If I have accomplished my purpose, then the book should be found usable both as a text for students and as a source of refer- ence for the more advanced mathematician.

I have tried to keep to a minimum the amount of new and unusual terminology and notation. In the few pI aces where my nomenclature differs from that in the existing literature of meas- ure theory, I was motivated by an attempt to harmonize with the usage of other parts of mathematics.

There are, for instance, sound algebraic reasons for using the terms "lattice" and "ring" for certain classes of sets-reasons which are more cogent than the similarities that caused Hausdorff to use "ring" and "field.

At the end of almost every section there is a set of exercises which appear sometimes as questions but more usually as asser- tions that the reader is invited to prove.

They constitute an integral part of the book; among them appear not only most of the examples and counter examples necessary for understanding the theory, but also definitions of new concepts and, occasionally, entire theories that not long aga were still subjects of research. It might appear inconsistent that, in the text, many elementary notions are treated in great detail, while,in the exercises,some quite refined and profound matters topological spaces, transfinite num- bers, Banach spaces, etc.

The mate- rial is arranged, however, so that when a beginning student comes to an exercise which uses terms not defined in this book he may simply omit it without loss of continuity. The more advanced reader, on the other hand, might be pleased at the interplay between measure theory and other parts of mathematics which it is the purpose of such exercises to exhibit.

The symbol I is used throughout the entire book in place of such phrases as "Q. At the end of the book there is a short list of references and a bibliography.

I make no claims of completeness for these lists. Their purpose is sometimes to mention background reading, rarely in cases where the history of the subject is not too well known to give credit for original discoveries, and most often to indicate directions for further study.

A symbol such as u. I am very much indebted to D. Blackwell, J. Doob, W. Gottschalk, L. Nachbin, B. Pettis, and, especially, to J. Oxtoby for their critical reading of the manuscript and their many valuable suggestions for improvements.

The result of 3. The condition in The example Set inclusion. Unions and interseetions. Limits, complements, and differences 16 4. Rings and algebras. Generated rings and q-rings. Monotone classes. Measure on rings. Measure on intervals. Properties of measures 37 Outer measures 41 Measurable sets. Properties of induced measures. Extension, completion, and approximation. Inner measures 58 15 Lebesgue measure. Measure spaces. Combinations of measurable functions 80 Sequences of measurable functions 84 Pointwise convergence.

Convergence in measure. Sequences of integrable simple functions 98 Integrable functions. Sequences of integrable functions. Signed measures. Hahn and Jordan decompositions Absolute continuity. The Radon-Nikodym theorem Cartesian products. Seetions Finite dimensional product spaces Measurable transformations.

Measure rings. Function spaces. Set functions and point functions. Series of indeJ? The law of large numbers. Conditional probabilities and expectations Measures on product spaces. Topologicallemmas Borel sets and Baire sets Regular measures Generation of Borel measures Regular contents Classes of continuous functions 24D Linear functionals.

Full subgroups Measurable groups. Topology in terms of measure Weil topology. Quotient groups. Specifically it is assumed that the reader is familiar with the concepts and results listed in 1 - 7 below. In this notation and In Chapter VIII the concept of metric space is used, together with such related concepts as completeness and separability for metric spaces, and uniform continuity of functions on metric spaces.

In Chapter VIII use is made also of such slightly more sophisticated concepts of real analysis as one-sided continuity. In general, each chapter makes free use of all preceding chap- ters; the only major exception to this is that Chapter IX is not needed for the last three chapters. In Chapters X, XI, and XII systematic use is made of many of the concepts and results of point set topology and the elements of topological group theory. We append below a list of all the relevant definitions and theorems.

The purpose of this list is not to serve as a text on topology, but Ca to tell the expert exactly [SEC. Topological Spaces A topological space is a set X and a dass of subsets of X, called the open sets of X, such that the dass contains 0 and X and is dosed under the formation of finite intersections and arbitrary i. The topo- logical space Xis discrete if every subset of X is open, or, equiva- lently, if every one-point subset of X is open. A set E is closed if X - Eis open.

The dass of dosed sets contains 0 and X and is dosed under the formation of finite unions and arbitrary inter- seetions. A subset Y of a topological space becomes a topological space a subspace of X in the relative topology if exactly those subsets of Y are called open which may be obtained by intersecting an open subset of X with Y. A neighborhood of a point x in X tor of a subset E of X] is an open set containing x tor an open set containing E]. A base is a dass B of open sets such that, for every x in X and every neighborhood U of x, there exists a set B in B such that x e B c U.

The topology of the realline is determined by the requirement that the dass of all open intervals be a base. A subbase is a dass of sets, the dass of all finite inter- seetions of which is a base. Aspace Xis separable if it has a countable base.

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