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Título : Measure Theory and Probability Theory Tipo de documento: documento electrónico Autores: Krishna B. Athreya ; SpringerLink (Online service) ; Lahiri, Soumendra N Editorial: New York, NY : Springer New York Fecha de publicación: 2006 Colección: Springer Texts in Statistics, ISSN 1431-875X Número de páginas: XVIII, 619 p Il.: online resource ISBN/ISSN/DL: 978-0-387-35434-7 Idioma : Inglés (eng) Palabras clave: Mathematics Computers Mathematical analysis Analysis (Mathematics) Measure theory Operations research Management science Probabilities Statistics Probability Theory and Stochastic Processes of Computation Integration Statistical Methods Research, Science Clasificación: 51 Matemáticas Resumen: This is a graduate level textbook on measure theory and probability theory. The book can be used as a text for a two semester sequence of courses in measure theory and probability theory, with an option to include supplemental material on stochastic processes and special topics. It is intended primarily for first year Ph.D. students in mathematics and statistics although mathematically advanced students from engineering and economics would also find the book useful. Prerequisites are kept to the minimal level of an understanding of basic real analysis concepts such as limits, continuity, differentiability, Riemann integration, and convergence of sequences and series. A review of this material is included in the appendix. The book starts with an informal introduction that provides some heuristics into the abstract concepts of measure and integration theory, which are then rigorously developed. The first part of the book can be used for a standard real analysis course for both mathematics and statistics Ph.D. students as it provides full coverage of topics such as the construction of Lebesgue-Stieltjes measures on real line and Euclidean spaces, the basic convergence theorems, L^p spaces, signed measures, Radon-Nikodym theorem, Lebesgue's decomposition theorem and the fundamental theorem of Lebesgue integration on R, product spaces and product measures, and Fubini-Tonelli theorems. It also provides an elementary introduction to Banach and Hilbert spaces, convolutions, Fourier series and Fourier and Plancherel transforms. Thus part I would be particularly useful for students in a typical Statistics Ph.D. program if a separate course on real analysis is not a standard requirement. Part II (chapters 6-13) provides full coverage of standard graduate level probability theory. It starts with Kolmogorov's probability model and Kolmogorov's existence theorem. It then treats thoroughly the laws of large numbers including renewal theory and ergodic theorems with applications and then weak convergence of probability distributions, characteristic functions, the Levy-Cramer continuity theorem and the central limit theorem as well as stable laws. It ends with conditional expectations and conditional probability, and an introduction to the theory of discrete time martingales. Part III (chapters 14-18) provides a modest coverage of discrete time Markov chains with countable and general state spaces, MCMC, continuous time discrete space jump Markov processes, Brownian motion, mixing sequences, bootstrap methods, and branching processes. It could be used for a topics/seminar course or as an introduction to stochastic processes. Krishna B. Athreya is a professor at the departments of mathematics and statistics and a Distinguished Professor in the College of Liberal Arts and Sciences at the Iowa State University. He has been a faculty member at University of Wisconsin, Madison; Indian Institute of Science, Bangalore; Cornell University; and has held visiting appointments in Scandinavia and Australia. He is a fellow of the Institute of Mathematical Statistics USA; a fellow of the Indian Academy of Sciences, Bangalore; an elected member of the International Statistical Institute; and serves on the editorial board of several journals in probability and statistics. Soumendra N. Lahiri is a professor at the department of statistics at the Iowa State University. He is a fellow of the Institute of Mathematical Statistics, a fellow of the American Statistical Association, and an elected member of the International Statistical Institute Nota de contenido: Measures and Integration: An Informal Introduction -- Measures -- Integration -- Lp-Spaces -- Differentiation -- Product Measures, Convolutions, and Transforms -- Probability Spaces -- Independence -- Laws of Large Numbers -- Convergence in Distribution -- Characteristic Functions -- Central Limit Theorems -- Conditional Expectation and Conditional Probability -- Discrete Parameter Martingales -- Markov Chains and MCMC -- Stochastic Processes -- Limit Theorems for Dependent Processes -- The Bootstrap -- Branching Processes En línea: http://dx.doi.org/10.1007/978-0-387-35434-7 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=34819 Measure Theory and Probability Theory [documento electrónico] / Krishna B. Athreya ; SpringerLink (Online service) ; Lahiri, Soumendra N . - New York, NY : Springer New York, 2006 . - XVIII, 619 p : online resource. - (Springer Texts in Statistics, ISSN 1431-875X) .
ISBN : 978-0-387-35434-7
Idioma : Inglés (eng)
Palabras clave: Mathematics Computers Mathematical analysis Analysis (Mathematics) Measure theory Operations research Management science Probabilities Statistics Probability Theory and Stochastic Processes of Computation Integration Statistical Methods Research, Science Clasificación: 51 Matemáticas Resumen: This is a graduate level textbook on measure theory and probability theory. The book can be used as a text for a two semester sequence of courses in measure theory and probability theory, with an option to include supplemental material on stochastic processes and special topics. It is intended primarily for first year Ph.D. students in mathematics and statistics although mathematically advanced students from engineering and economics would also find the book useful. Prerequisites are kept to the minimal level of an understanding of basic real analysis concepts such as limits, continuity, differentiability, Riemann integration, and convergence of sequences and series. A review of this material is included in the appendix. The book starts with an informal introduction that provides some heuristics into the abstract concepts of measure and integration theory, which are then rigorously developed. The first part of the book can be used for a standard real analysis course for both mathematics and statistics Ph.D. students as it provides full coverage of topics such as the construction of Lebesgue-Stieltjes measures on real line and Euclidean spaces, the basic convergence theorems, L^p spaces, signed measures, Radon-Nikodym theorem, Lebesgue's decomposition theorem and the fundamental theorem of Lebesgue integration on R, product spaces and product measures, and Fubini-Tonelli theorems. It also provides an elementary introduction to Banach and Hilbert spaces, convolutions, Fourier series and Fourier and Plancherel transforms. Thus part I would be particularly useful for students in a typical Statistics Ph.D. program if a separate course on real analysis is not a standard requirement. Part II (chapters 6-13) provides full coverage of standard graduate level probability theory. It starts with Kolmogorov's probability model and Kolmogorov's existence theorem. It then treats thoroughly the laws of large numbers including renewal theory and ergodic theorems with applications and then weak convergence of probability distributions, characteristic functions, the Levy-Cramer continuity theorem and the central limit theorem as well as stable laws. It ends with conditional expectations and conditional probability, and an introduction to the theory of discrete time martingales. Part III (chapters 14-18) provides a modest coverage of discrete time Markov chains with countable and general state spaces, MCMC, continuous time discrete space jump Markov processes, Brownian motion, mixing sequences, bootstrap methods, and branching processes. It could be used for a topics/seminar course or as an introduction to stochastic processes. Krishna B. Athreya is a professor at the departments of mathematics and statistics and a Distinguished Professor in the College of Liberal Arts and Sciences at the Iowa State University. He has been a faculty member at University of Wisconsin, Madison; Indian Institute of Science, Bangalore; Cornell University; and has held visiting appointments in Scandinavia and Australia. He is a fellow of the Institute of Mathematical Statistics USA; a fellow of the Indian Academy of Sciences, Bangalore; an elected member of the International Statistical Institute; and serves on the editorial board of several journals in probability and statistics. Soumendra N. Lahiri is a professor at the department of statistics at the Iowa State University. He is a fellow of the Institute of Mathematical Statistics, a fellow of the American Statistical Association, and an elected member of the International Statistical Institute Nota de contenido: Measures and Integration: An Informal Introduction -- Measures -- Integration -- Lp-Spaces -- Differentiation -- Product Measures, Convolutions, and Transforms -- Probability Spaces -- Independence -- Laws of Large Numbers -- Convergence in Distribution -- Characteristic Functions -- Central Limit Theorems -- Conditional Expectation and Conditional Probability -- Discrete Parameter Martingales -- Markov Chains and MCMC -- Stochastic Processes -- Limit Theorems for Dependent Processes -- The Bootstrap -- Branching Processes En línea: http://dx.doi.org/10.1007/978-0-387-35434-7 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=34819 Ejemplares
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Título : Measure Theory Tipo de documento: documento electrónico Autores: Bogachev, Vladimir I ; SpringerLink (Online service) Editorial: Berlin, Heidelberg : Springer Berlin Heidelberg Fecha de publicación: 2007 Número de páginas: XXX, 1075 p Il.: online resource ISBN/ISSN/DL: 978-3-540-34514-5 Idioma : Inglés (eng) Palabras clave: Mathematics Mathematical analysis Analysis (Mathematics) Functional Measure theory Probabilities and Integration Probability Theory Stochastic Processes Clasificación: 51 Matemáticas Resumen: Measure theory is a classical area of mathematics that continues intensive development and has fruitful connections with most other fields of mathematics as well as important applications in physics. This book gives a systematic presentation of modern measure theory as it has developed over the past century and offers three levels of presentation: a standard university graduate course, an advanced study containing some complements to the basic course (the material of this level corresponds to a variety of special courses), and, finally, more specialized topics partly covered by more than 850 exercises. Bibliographical and historical comments and an extensive bibliography with 2000 works covering more than a century are provided. Volume 1 is devoted to the classical theory of measure and integral. Whereas the first volume presents the ideas that go back mainly to Lebesgue, the second volume is to a large extent the result of the later development up to the recent years. The central subjects of Volume 2 are: transformations of measures, conditional measures, and weak convergence of measures. These topics are closely interwoven and form the heart of modern measure theory. The target readership includes graduate students interested in deeper knowledge of measure theory, instructors of courses in measure and integration theory, and researchers in all fields of mathematics. The book may serve as a source for many advanced courses or as a reference Nota de contenido: Constructions and extensions of measures -- The Lebesgue integral -- Operations on measures and functions -- The spaces Lp and spaces of measures -- Connections between the integral and derivative -- Borel, Baire and Souslin sets -- Measures on topological spaces -- Weak convergence of measure -- Transformations of measures and isomorphisms -- Conditional measures and conditional En línea: http://dx.doi.org/10.1007/978-3-540-34514-5 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=34616 Measure Theory [documento electrónico] / Bogachev, Vladimir I ; SpringerLink (Online service) . - Berlin, Heidelberg : Springer Berlin Heidelberg, 2007 . - XXX, 1075 p : online resource.
ISBN : 978-3-540-34514-5
Idioma : Inglés (eng)
Palabras clave: Mathematics Mathematical analysis Analysis (Mathematics) Functional Measure theory Probabilities and Integration Probability Theory Stochastic Processes Clasificación: 51 Matemáticas Resumen: Measure theory is a classical area of mathematics that continues intensive development and has fruitful connections with most other fields of mathematics as well as important applications in physics. This book gives a systematic presentation of modern measure theory as it has developed over the past century and offers three levels of presentation: a standard university graduate course, an advanced study containing some complements to the basic course (the material of this level corresponds to a variety of special courses), and, finally, more specialized topics partly covered by more than 850 exercises. Bibliographical and historical comments and an extensive bibliography with 2000 works covering more than a century are provided. Volume 1 is devoted to the classical theory of measure and integral. Whereas the first volume presents the ideas that go back mainly to Lebesgue, the second volume is to a large extent the result of the later development up to the recent years. The central subjects of Volume 2 are: transformations of measures, conditional measures, and weak convergence of measures. These topics are closely interwoven and form the heart of modern measure theory. The target readership includes graduate students interested in deeper knowledge of measure theory, instructors of courses in measure and integration theory, and researchers in all fields of mathematics. The book may serve as a source for many advanced courses or as a reference Nota de contenido: Constructions and extensions of measures -- The Lebesgue integral -- Operations on measures and functions -- The spaces Lp and spaces of measures -- Connections between the integral and derivative -- Borel, Baire and Souslin sets -- Measures on topological spaces -- Weak convergence of measure -- Transformations of measures and isomorphisms -- Conditional measures and conditional En línea: http://dx.doi.org/10.1007/978-3-540-34514-5 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=34616 Ejemplares
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Título : Measure and Integration : Publications 1997-2011 Tipo de documento: documento electrónico Autores: König, Heinz ; SpringerLink (Online service) Editorial: Basel : Springer Basel Fecha de publicación: 2012 Otro editor: Imprint: Birkhäuser Número de páginas: XII, 512 p Il.: online resource ISBN/ISSN/DL: 978-3-0348-0382-3 Idioma : Inglés (eng) Palabras clave: Mathematics Measure theory and Integration Clasificación: 51 Matemáticas Resumen: This volume presents a collection of twenty-five of Heinz König’s recent and most influential works. Connecting to his book of 1997 “Measure and Integration”, the author has developed a consistent new version of measure theory over the past years. For the first time, his publications are collected here in one single volume. Key features include: - A first-time, original and entirely uniform treatment of abstract and topological measure theory - The introduction of the inner • and outer • premeasures and their extension to unique maximal measures - A simplification of the procedure formerly described in Chapter II of the author’s previous book - The creation of new “envelopes” for the initial set function (to replace the traditional Carathéodory outer measures), which lead to much simpler and more explicit treatment - The formation of products, a unified Daniell-Stone-Riesz representation theorem, and projective limits, which allows to obtain the Kolmogorov type projective limit theorem for even huge domains far beyond the countably determined ones - The incorporation of non-sequential and of inner regular versions, which leads to much more comprehensive results - Significant applications to stochastic processes. “Measure and Integration: Publications 1997–2011” will appeal to both researchers and advanced graduate students in the fields of measure and integration and probabilistic measure theory Nota de contenido: Image measures and the so-called image measure catastrophe -- The product theory for inner premeasures -- Measure and Integration: Mutual generation of outer and inner premeasures -- Measure and Integration: Integral representations of isotone functionals -- Measure and Integration: Comparison of old and new procedures -- What are signed contents and measures?- Upper envelopes of inner premeasures -- On the inner Daniell-Stone and Riesz representation theorems -- Sublinear functionals and conical measures -- Measure and Integration: An attempt at unified systematization -- New facts around the Choquet integral -- The (sub/super)additivity assertion of Choquet -- Projective limits via inner premeasures and the trueWiener measure -- Stochastic processes in terms of inner premeasures -- New versions of the Radon-Nikodým theorem -- The Lebesgue decomposition theorem for arbitrary contents -- The new maximal measures for stochastic processes -- Stochastic processes on the basis of new measure theory -- New versions of the Daniell-Stone-Riesz representation theorem -- Measure and Integral: New foundations after one hundred years -- Fubini-Tonelli theorems on the basis of inner and outer premeasures -- Measure and Integration: Characterization of the new maximal contents and measures -- Notes on the projective limit theorem of Kolmogorov -- Measure and Integration: The basic extension theorems -- Measure Theory: Transplantation theorems for inner premeasures. En línea: http://dx.doi.org/10.1007/978-3-0348-0382-3 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=32884 Measure and Integration : Publications 1997-2011 [documento electrónico] / König, Heinz ; SpringerLink (Online service) . - Basel : Springer Basel : Imprint: Birkhäuser, 2012 . - XII, 512 p : online resource.
ISBN : 978-3-0348-0382-3
Idioma : Inglés (eng)
Palabras clave: Mathematics Measure theory and Integration Clasificación: 51 Matemáticas Resumen: This volume presents a collection of twenty-five of Heinz König’s recent and most influential works. Connecting to his book of 1997 “Measure and Integration”, the author has developed a consistent new version of measure theory over the past years. For the first time, his publications are collected here in one single volume. Key features include: - A first-time, original and entirely uniform treatment of abstract and topological measure theory - The introduction of the inner • and outer • premeasures and their extension to unique maximal measures - A simplification of the procedure formerly described in Chapter II of the author’s previous book - The creation of new “envelopes” for the initial set function (to replace the traditional Carathéodory outer measures), which lead to much simpler and more explicit treatment - The formation of products, a unified Daniell-Stone-Riesz representation theorem, and projective limits, which allows to obtain the Kolmogorov type projective limit theorem for even huge domains far beyond the countably determined ones - The incorporation of non-sequential and of inner regular versions, which leads to much more comprehensive results - Significant applications to stochastic processes. “Measure and Integration: Publications 1997–2011” will appeal to both researchers and advanced graduate students in the fields of measure and integration and probabilistic measure theory Nota de contenido: Image measures and the so-called image measure catastrophe -- The product theory for inner premeasures -- Measure and Integration: Mutual generation of outer and inner premeasures -- Measure and Integration: Integral representations of isotone functionals -- Measure and Integration: Comparison of old and new procedures -- What are signed contents and measures?- Upper envelopes of inner premeasures -- On the inner Daniell-Stone and Riesz representation theorems -- Sublinear functionals and conical measures -- Measure and Integration: An attempt at unified systematization -- New facts around the Choquet integral -- The (sub/super)additivity assertion of Choquet -- Projective limits via inner premeasures and the trueWiener measure -- Stochastic processes in terms of inner premeasures -- New versions of the Radon-Nikodým theorem -- The Lebesgue decomposition theorem for arbitrary contents -- The new maximal measures for stochastic processes -- Stochastic processes on the basis of new measure theory -- New versions of the Daniell-Stone-Riesz representation theorem -- Measure and Integral: New foundations after one hundred years -- Fubini-Tonelli theorems on the basis of inner and outer premeasures -- Measure and Integration: Characterization of the new maximal contents and measures -- Notes on the projective limit theorem of Kolmogorov -- Measure and Integration: The basic extension theorems -- Measure Theory: Transplantation theorems for inner premeasures. En línea: http://dx.doi.org/10.1007/978-3-0348-0382-3 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=32884 Ejemplares
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Título : Measure, Integral, Derivative : A Course on Lebesgue's Theory Tipo de documento: documento electrónico Autores: Ovchinnikov, Sergei ; SpringerLink (Online service) Editorial: New York, NY : Springer New York Fecha de publicación: 2013 Otro editor: Imprint: Springer Colección: Universitext, ISSN 0172-5939 Número de páginas: X, 146 p. 16 illus Il.: online resource ISBN/ISSN/DL: 978-1-4614-7196-7 Idioma : Inglés (eng) Palabras clave: Mathematics Mathematical analysis Analysis (Mathematics) Measure theory Functions of real variables and Integration Real Clasificación: 51 Matemáticas Resumen: This classroom-tested text is intended for a one-semester course in Lebesgue’s theory. With over 180 exercises, the text takes an elementary approach, making it easily accessible to both upper-undergraduate- and lower-graduate-level students. The three main topics presented are measure, integration, and differentiation, and the only prerequisite is a course in elementary real analysis. In order to keep the book self-contained, an introductory chapter is included with the intent to fill the gap between what the student may have learned before and what is required to fully understand the consequent text. Proofs of difficult results, such as the differentiability property of functions of bounded variations, are dissected into small steps in order to be accessible to students. With the exception of a few simple statements, all results are proven in the text. The presentation is elementary, where s-algebras are not used in the text on measure theory and Dini’s derivatives are not used in the chapter on differentiation. However, all the main results of Lebesgue’s theory are found in the book Nota de contenido: 1 Preliminaries -- 2 Lebesgue Measure -- 3 Lebesgue Integration -- 4 Differentiation and Integration -- A Measure and Integral over Unbounded Sets -- Index En línea: http://dx.doi.org/10.1007/978-1-4614-7196-7 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=32341 Measure, Integral, Derivative : A Course on Lebesgue's Theory [documento electrónico] / Ovchinnikov, Sergei ; SpringerLink (Online service) . - New York, NY : Springer New York : Imprint: Springer, 2013 . - X, 146 p. 16 illus : online resource. - (Universitext, ISSN 0172-5939) .
ISBN : 978-1-4614-7196-7
Idioma : Inglés (eng)
Palabras clave: Mathematics Mathematical analysis Analysis (Mathematics) Measure theory Functions of real variables and Integration Real Clasificación: 51 Matemáticas Resumen: This classroom-tested text is intended for a one-semester course in Lebesgue’s theory. With over 180 exercises, the text takes an elementary approach, making it easily accessible to both upper-undergraduate- and lower-graduate-level students. The three main topics presented are measure, integration, and differentiation, and the only prerequisite is a course in elementary real analysis. In order to keep the book self-contained, an introductory chapter is included with the intent to fill the gap between what the student may have learned before and what is required to fully understand the consequent text. Proofs of difficult results, such as the differentiability property of functions of bounded variations, are dissected into small steps in order to be accessible to students. With the exception of a few simple statements, all results are proven in the text. The presentation is elementary, where s-algebras are not used in the text on measure theory and Dini’s derivatives are not used in the chapter on differentiation. However, all the main results of Lebesgue’s theory are found in the book Nota de contenido: 1 Preliminaries -- 2 Lebesgue Measure -- 3 Lebesgue Integration -- 4 Differentiation and Integration -- A Measure and Integral over Unbounded Sets -- Index En línea: http://dx.doi.org/10.1007/978-1-4614-7196-7 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=32341 Ejemplares
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Título : Measure Theory : Second Edition Tipo de documento: documento electrónico Autores: Cohn, Donald L ; SpringerLink (Online service) Editorial: New York, NY : Springer New York Fecha de publicación: 2013 Otro editor: Imprint: Birkhäuser Colección: Birkhäuser Advanced Texts Basler Lehrbücher, ISSN 1019-6242 Número de páginas: XXI, 457 p Il.: online resource ISBN/ISSN/DL: 978-1-4614-6956-8 Idioma : Inglés (eng) Palabras clave: Mathematics Mathematical analysis Analysis (Mathematics) Measure theory Probabilities and Integration Probability Theory Stochastic Processes Clasificación: 51 Matemáticas Resumen: Intended as a self-contained introduction to measure theory, this textbook also includes a comprehensive treatment of integration on locally compact Hausdorff spaces, the analytic and Borel subsets of Polish spaces, and Haar measures on locally compact groups. This second edition includes a chapter on measure-theoretic probability theory, plus brief treatments of the Banach-Tarski paradox, the Henstock-Kurzweil integral, the Daniell integral, and the existence of liftings. Measure Theory provides a solid background for study in both functional analysis and probability theory and is an excellent resource for advanced undergraduate and graduate students in mathematics. The prerequisites for this book are basic courses in point-set topology and in analysis, and the appendices present a thorough review of essential background material. The author aims to present a straightforward treatment of the part of measure theory necessary for analysis and probability' assuming only basic knowledge of analysis and topology...Each chapter includes numerous well-chosen exercises, varying from very routine practice problems to important extensions and developments of the theory; for the difficult ones there are helpful hints. It is the reviewer's opinion that the author has succeeded in his aim. In spite of its lack of new results, the selection and presentation of materials makes this a useful book for an introduction to measure and integration theory. —Mathematical Reviews (Review of the First Edition) The book is a comprehensive and clearly written textbook on measure and integration...The book contains appendices on set theory, algebra, calculus and topology in Euclidean spaces, topological and metric spaces, and the Bochner integral. Each section of the book contains a number of exercises. —zbMATH (Review of the First Edition) Nota de contenido: 1. Measures -- Algebras and sigma-algebras -- Measures -- Outer measures -- Lebesgue measure -- Completeness and regularity -- Dynkin classes -- 2. Functions and Integrals -- Measurable functions -- Properties that hold almost everywhere -- The integral -- Limit theorems -- The Riemann integral -- Measurable functions again, complex-valued functions, and image measures -- 3. Convergence -- Modes of Convergence -- Normed spaces -- Definition of L^p and L^p -- Properties of L^p and L-p -- Dual spaces -- 4. Signed and Complex Measures -- Signed and complex measures -- Absolute continuity -- Singularity -- Functions of bounded variation -- The duals of the L^p spaces -- 5. Product Measures -- Constructions -- Fubini’s theorem -- Applications -- 6. Differentiation -- Change of variable in R^d -- Differentiation of measures -- Differentiation of functions -- 7. Measures on Locally Compact Spaces -- Locally compact spaces -- The Riesz representation theorem -- Signed and complex measures; duality -- Additional properties of regular measures -- The u^*-measurable sets and the dual of L^1 -- Products of locally compact spaces -- 8. Polish Spaces and Analytic Sets -- Polish spaces -- Analytic sets -- The separation theorem and its consequences -- The measurability of analytic sets -- Cross sections -- Standard, analytic, Lusin, and Souslin spaces -- 9. Haar Measure -- Topological groups -- The existence and uniqueness of Haar measure -- The algebras L^1 (G) and M (G) -- Appendices -- A. Notation and set theory -- B. Algebra -- C. Calculus and topology in R^d -- D. Topological spaces and metric spaces -- E. The Bochner integral -- F Liftings -- G The Banach-Tarski paradox -- H The Henstock-Kurzweil and McShane integralsBibliography -- Index of notation -- Index En línea: http://dx.doi.org/10.1007/978-1-4614-6956-8 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=32326 Measure Theory : Second Edition [documento electrónico] / Cohn, Donald L ; SpringerLink (Online service) . - New York, NY : Springer New York : Imprint: Birkhäuser, 2013 . - XXI, 457 p : online resource. - (Birkhäuser Advanced Texts Basler Lehrbücher, ISSN 1019-6242) .
ISBN : 978-1-4614-6956-8
Idioma : Inglés (eng)
Palabras clave: Mathematics Mathematical analysis Analysis (Mathematics) Measure theory Probabilities and Integration Probability Theory Stochastic Processes Clasificación: 51 Matemáticas Resumen: Intended as a self-contained introduction to measure theory, this textbook also includes a comprehensive treatment of integration on locally compact Hausdorff spaces, the analytic and Borel subsets of Polish spaces, and Haar measures on locally compact groups. This second edition includes a chapter on measure-theoretic probability theory, plus brief treatments of the Banach-Tarski paradox, the Henstock-Kurzweil integral, the Daniell integral, and the existence of liftings. Measure Theory provides a solid background for study in both functional analysis and probability theory and is an excellent resource for advanced undergraduate and graduate students in mathematics. The prerequisites for this book are basic courses in point-set topology and in analysis, and the appendices present a thorough review of essential background material. The author aims to present a straightforward treatment of the part of measure theory necessary for analysis and probability' assuming only basic knowledge of analysis and topology...Each chapter includes numerous well-chosen exercises, varying from very routine practice problems to important extensions and developments of the theory; for the difficult ones there are helpful hints. It is the reviewer's opinion that the author has succeeded in his aim. In spite of its lack of new results, the selection and presentation of materials makes this a useful book for an introduction to measure and integration theory. —Mathematical Reviews (Review of the First Edition) The book is a comprehensive and clearly written textbook on measure and integration...The book contains appendices on set theory, algebra, calculus and topology in Euclidean spaces, topological and metric spaces, and the Bochner integral. Each section of the book contains a number of exercises. —zbMATH (Review of the First Edition) Nota de contenido: 1. Measures -- Algebras and sigma-algebras -- Measures -- Outer measures -- Lebesgue measure -- Completeness and regularity -- Dynkin classes -- 2. Functions and Integrals -- Measurable functions -- Properties that hold almost everywhere -- The integral -- Limit theorems -- The Riemann integral -- Measurable functions again, complex-valued functions, and image measures -- 3. Convergence -- Modes of Convergence -- Normed spaces -- Definition of L^p and L^p -- Properties of L^p and L-p -- Dual spaces -- 4. Signed and Complex Measures -- Signed and complex measures -- Absolute continuity -- Singularity -- Functions of bounded variation -- The duals of the L^p spaces -- 5. Product Measures -- Constructions -- Fubini’s theorem -- Applications -- 6. Differentiation -- Change of variable in R^d -- Differentiation of measures -- Differentiation of functions -- 7. Measures on Locally Compact Spaces -- Locally compact spaces -- The Riesz representation theorem -- Signed and complex measures; duality -- Additional properties of regular measures -- The u^*-measurable sets and the dual of L^1 -- Products of locally compact spaces -- 8. Polish Spaces and Analytic Sets -- Polish spaces -- Analytic sets -- The separation theorem and its consequences -- The measurability of analytic sets -- Cross sections -- Standard, analytic, Lusin, and Souslin spaces -- 9. Haar Measure -- Topological groups -- The existence and uniqueness of Haar measure -- The algebras L^1 (G) and M (G) -- Appendices -- A. Notation and set theory -- B. Algebra -- C. Calculus and topology in R^d -- D. Topological spaces and metric spaces -- E. The Bochner integral -- F Liftings -- G The Banach-Tarski paradox -- H The Henstock-Kurzweil and McShane integralsBibliography -- Index of notation -- Index En línea: http://dx.doi.org/10.1007/978-1-4614-6956-8 Link: https://biblioteca.cunef.edu/gestion/catalogo/index.php?lvl=notice_display&id=32326 Ejemplares
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