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| Author | : René L. Schilling |
| Publisher | : Cambridge University Press |
| Total Pages | : 497 |
| Release | : 2017-04-03 |
| Genre | : Mathematics |
| ISBN | : 1316620247 |
A concise, elementary introduction to measure and integration theory, requiring few prerequisites as theory is developed quickly and simply.
| Author | : René L. Schilling |
| Publisher | : Cambridge University Press |
| Total Pages | : 404 |
| Release | : 2005-11-10 |
| Genre | : Mathematics |
| ISBN | : 9780521850155 |
This book, first published in 2005, introduces measure and integration theory as it is needed in many parts of analysis and probability.
| Author | : René L. Schilling |
| Publisher | : Cambridge University Press |
| Total Pages | : 431 |
| Release | : 2021-06-17 |
| Genre | : Mathematics |
| ISBN | : 1009020390 |
Often it is more instructive to know 'what can go wrong' and to understand 'why a result fails' than to plod through yet another piece of theory. In this text, the authors gather more than 300 counterexamples - some of them both surprising and amusing - showing the limitations, hidden traps and pitfalls of measure and integration. Many examples are put into context, explaining relevant parts of the theory, and pointing out further reading. The text starts with a self-contained, non-technical overview on the fundamentals of measure and integration. A companion to the successful undergraduate textbook Measures, Integrals and Martingales, it is accessible to advanced undergraduate students, requiring only modest prerequisites. More specialized concepts are summarized at the beginning of each chapter, allowing for self-study as well as supplementary reading for any course covering measures and integrals. For researchers, it provides ample examples and warnings as to the limitations of general measure theory. This book forms a sister volume to René Schilling's other book Measures, Integrals and Martingales (www.cambridge.org/9781316620243).
| Author | : Marek Capinski |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 229 |
| Release | : 2013-06-29 |
| Genre | : Mathematics |
| ISBN | : 1447136314 |
This very well written and accessible book emphasizes the reasons for studying measure theory, which is the foundation of much of probability. By focusing on measure, many illustrative examples and applications, including a thorough discussion of standard probability distributions and densities, are opened. The book also includes many problems and their fully worked solutions.
| Author | : Rene L. Schilling |
| Publisher | : |
| Total Pages | : 0 |
| Release | : 2017 |
| Genre | : |
| ISBN | : 9781108161275 |
| Author | : René L. Schilling |
| Publisher | : Cambridge University Press |
| Total Pages | : 48 |
| Release | : 2005-11-10 |
| Genre | : Mathematics |
| ISBN | : 1139446533 |
This book, first published in 2005, introduces measure and integration theory as it is needed in many parts of analysis and probability theory. The basic theory - measures, integrals, convergence theorems, Lp-spaces and multiple integrals - is explored in the first part of the book. The second part then uses the notion of martingales to develop the theory further, covering topics such as Jacobi's generalized transformation Theorem, the Radon-Nikodym theorem, Hardy-Littlewood maximal functions or general Fourier series. Undergraduate calculus and an introductory course on rigorous analysis are the only essential prerequisites, making this text suitable for both lecture courses and for self-study. Numerous illustrations and exercises are included and these are not merely drill problems but are there to consolidate what has already been learnt and to discover variants, sideways and extensions to the main material. Hints and solutions can be found on the author's website, which can be reached from www.cambridge.org/9780521615259. This book forms a sister volume to René Schilling's other book Counterexamples in Measure and Integration (www.cambridge.org/9781009001625).
| Author | : P. E. Kopp |
| Publisher | : Cambridge University Press |
| Total Pages | : 0 |
| Release | : 2008-11-20 |
| Genre | : Mathematics |
| ISBN | : 9780521090339 |
This book provides an introduction to the rapidly expanding theory of stochastic integration and martingales. The treatment is close to that developed by the French school of probabilists, but is more elementary than other texts. The presentation is abstract, but largely self-contained and Dr Kopp makes fewer demands on the reader's background in probability theory than is usual. He gives a fairly full discussion of the measure theory and functional analysis needed for martingale theory, and describes the role of Brownian motion and the Poisson process as paradigm examples in the construction of abstract stochastic integrals. An appendix provides the reader with a glimpse of very recent developments in non-commutative integration theory which are of considerable importance in quantum mechanics. Thus equipped, the reader will have the necessary background to understand research in stochastic analysis. As a textbook, this account will be ideally suited to beginning graduate students in probability theory, and indeed it has evolved from such courses given at Hull University. It should also be of interest to pure mathematicians looking for a careful, yet concise introduction to martingale theory, and to physicists, engineers and economists who are finding that applications to their disciplines are becoming increasingly important.
| Author | : Sheldon Axler |
| Publisher | : Springer Nature |
| Total Pages | : 430 |
| Release | : 2019-11-29 |
| Genre | : Mathematics |
| ISBN | : 3030331431 |
This open access textbook welcomes students into the fundamental theory of measure, integration, and real analysis. Focusing on an accessible approach, Axler lays the foundations for further study by promoting a deep understanding of key results. Content is carefully curated to suit a single course, or two-semester sequence of courses, creating a versatile entry point for graduate studies in all areas of pure and applied mathematics. Motivated by a brief review of Riemann integration and its deficiencies, the text begins by immersing students in the concepts of measure and integration. Lebesgue measure and abstract measures are developed together, with each providing key insight into the main ideas of the other approach. Lebesgue integration links into results such as the Lebesgue Differentiation Theorem. The development of products of abstract measures leads to Lebesgue measure on Rn. Chapters on Banach spaces, Lp spaces, and Hilbert spaces showcase major results such as the Hahn–Banach Theorem, Hölder’s Inequality, and the Riesz Representation Theorem. An in-depth study of linear maps on Hilbert spaces culminates in the Spectral Theorem and Singular Value Decomposition for compact operators, with an optional interlude in real and complex measures. Building on the Hilbert space material, a chapter on Fourier analysis provides an invaluable introduction to Fourier series and the Fourier transform. The final chapter offers a taste of probability. Extensively class tested at multiple universities and written by an award-winning mathematical expositor, Measure, Integration & Real Analysis is an ideal resource for students at the start of their journey into graduate mathematics. A prerequisite of elementary undergraduate real analysis is assumed; students and instructors looking to reinforce these ideas will appreciate the electronic Supplement for Measure, Integration & Real Analysis that is freely available online. For errata and updates, visit https://measure.axler.net/
| Author | : James J Yeh |
| Publisher | : World Scientific Publishing Company |
| Total Pages | : 840 |
| Release | : 2014-06-11 |
| Genre | : Mathematics |
| ISBN | : 9814578568 |
This book presents a unified treatise of the theory of measure and integration. In the setting of a general measure space, every concept is defined precisely and every theorem is presented with a clear and complete proof with all the relevant details. Counter-examples are provided to show that certain conditions in the hypothesis of a theorem cannot be simply dropped. The dependence of a theorem on earlier theorems is explicitly indicated in the proof, not only to facilitate reading but also to delineate the structure of the theory. The precision and clarity of presentation make the book an ideal textbook for a graduate course in real analysis while the wealth of topics treated also make the book a valuable reference work for mathematicians.The book is also very helpful to graduate students in statistics and electrical engineering, two disciplines that apply measure theory.
| Author | : Jean-François Le Gall |
| Publisher | : Springer |
| Total Pages | : 282 |
| Release | : 2016-04-28 |
| Genre | : Mathematics |
| ISBN | : 3319310895 |
This book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Itô’s formula, the optional stopping theorem and Girsanov’s theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter. Since its invention by Itô, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested in such developments. Beginning graduate or advanced undergraduate students will benefit from this detailed approach to an essential area of probability theory. The emphasis is on concise and efficient presentation, without any concession to mathematical rigor. The material has been taught by the author for several years in graduate courses at two of the most prestigious French universities. The fact that proofs are given with full details makes the book particularly suitable for self-study. The numerous exercises help the reader to get acquainted with the tools of stochastic calculus.
