測度與范疇學(xué)

前言

This book has two main themes：the Baire category theorem as a method for proving existence。and the“duality”between me.~SUl'e and category. The category method iS illustrated by a variety of typical applications， and the analogy between measure and category iS explored in all of its ramifications.To this end,the elements of metric topology are reviewed and the principal properties of Lcbesgue measure are derived.It turns out that Lebesgue integration is not essential for present purposes-ltheRiemann integral is SUfIident.Concepts of general measure theory andtopology are introduced，but not iust for the sake of generality.Needlesstosay,theterm“category”refersalwaystoBairecategory；ithasnothingtodOwiththetermasitiSusedin homologicalalgebra.A knowiedge of calculus is presupposed,and some familiarity with the algebra of sets.The questions discussed are ones that lend themselves naturally to set-theoretical formulation.The book is intended as an introduction to this kind of analysis.It could be used to supplement a standard cOUrse in real analysis,as the basis for a seminar,or for inde. pendent study.It is primarily expository。but a few refinements of known results are included,notably Theorem 15.6 and Proposition 204.The references are not intended to be complete.Frequently a secondary source is cited where additional references may be found.The book iS a revised and expanded version of notes originaily prepared for a course of lectures givfn at Haverford College during the spring of 1957 under the auspiccs of the William Pyle Philips Fund. These，in turn，were based on the Earle Raymond Hedrick Lectures presented at the Summer Meeting of the Mathematical Association of America at Seattle,Washington。in August.1956.

內(nèi)容概要

　　This book has two main themes： the Baire category theorem as a method for proving existence, and the "duality" between measure and category. The category method is illustrated by a variety of typical applications, and the analogy between measure and category is explored in all of its ramifications. To this end, the elements of metric topology are reviewed and the principal properties of Lebesgue measure are derived. It turns out that Lebesgue integration is not essential for present purposes——the Riemann integral is sufficient. Concepts of general measure theory and topology are introduced, but not just for the sake of generality. Needless to say, the term "category" refers always to Baire category; it has nothing to do with the term as it is used in homological algebra

作者簡介

作者：(美國) 奧凱斯屠拜 (Oxtoby.J.C.)

書籍目錄

1．Measure and Category on the Line．，　Countable sets,sets offirst category，nullsets,the theorems ofCantor．Baire,and BOreI2．Liouville Numbers．　Algebraic and transcendental numbers,measure and category of the set of　Liouviile humbers3．Lcbesgue Measure in r-Space．’  Definitions and principal properties，measurable sets，the Lebesgue density theorem4．The Property of Baire　Its analogy to measurability，properties of regular open sets5．Non-Measurable Sets　Vitali sets,Bernstein sets，Ulam’s theorem,inaccessible cardinals,the con．　tinuum hypothesis6．The Banach-Mazur Game  Winning strategies,categoff and local category,indeterminate games7．Functions of First Class．　Oscillation,the limit of a sequence of continuous functions,Riemann integrability8．The Theorems of Lusin and Egoroff  Continuity of measurablc functions and of functimis having the property of Baire,uniform convergence on subsets　9．Metric and Topological Spaces  Definitions,complete and topologically complete spaces,the Baire categorytheorem　10．Examples of Metric Spaces　Uniform and integral metrics in the space of continuous functions,integrabl functions,pseudmetric spaces，the space of measurable sets11．Nowhere Differentiable Functions  Banach’S application of the category method12．The Theorem ofAlexandroff．　Remetrization of a G．subset，topologically complete subspaces13．Transforming Linear Sets into Nullsets’．　The space of automorphisms of an interval．effect of monotone substitution ORiemann integrability．nullsets equivalent to sets of first category14．Fubini’S Theorem　Measurability and measure of sections of plane measurable sets15．The Kuratowski．Ulam Theorem．，　Sections of plane sets having the property of Baire．product sets，reducibility tFubinis theorem by means of a product transformation16．The Banach Category Theorem．　Open sets of first category or measure zero,MontgomeryS lemma，the thecrums of Marczewski and Sikorski．cardinals of measure zerodecomposition into a nullset and a set of first category17．The Poincar6 Recurrence Theorem　Measure and category of the set of points recurrent under a nondissipativ transformation，application to dynamical systems18．Transitive Transformations　Existence of transitive automorphisms of the square,the category method19．The Sierpinski·Erd6s Duality Theorem．　Similarities between the classes of sets of measure zero and of first category，th principie of duality20．Examples of Duality．　Properties of Lusin sets and their duals，sets almost invariant under transformations that preserve nullsets or category21．The Extended Principle of Duality．　A counter example．product measures and product spaces，the zero-one law and its category analogue22．Category Measure Spaces　Spaces in which measure and category agree，topologies generated by lowe densities,the Lebesgue density topologySupplementary Notes and Remarks．References Supplementary References．Index

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