Introduction to Measure Theory and Integration

Author: Luigi Ambrosio
Publisher: Springer Science & Business Media
ISBN: 8876423869
Format: PDF
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This textbook collects the notes for an introductory course in measure theory and integration. The course was taught by the authors to undergraduate students of the Scuola Normale Superiore, in the years 2000-2011. The goal of the course was to present, in a quick but rigorous way, the modern point of view on measure theory and integration, putting Lebesgue's Euclidean space theory into a more general context and presenting the basic applications to Fourier series, calculus and real analysis. The text can also pave the way to more advanced courses in probability, stochastic processes or geometric measure theory. Prerequisites for the book are a basic knowledge of calculus in one and several variables, metric spaces and linear algebra. All results presented here, as well as their proofs, are classical. The authors claim some originality only in the presentation and in the choice of the exercises. Detailed solutions to the exercises are provided in the final part of the book.

A Course on Integration Theory

Author: Nicolas Lerner
Publisher: Springer
ISBN: 3034806949
Format: PDF, ePub, Mobi
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This textbook provides a detailed treatment of abstract integration theory, construction of the Lebesgue measure via the Riesz-Markov Theorem and also via the Carathéodory Theorem. It also includes some elementary properties of Hausdorff measures as well as the basic properties of spaces of integrable functions and standard theorems on integrals depending on a parameter. Integration on a product space, change of variables formulas as well as the construction and study of classical Cantor sets are treated in detail. Classical convolution inequalities, such as Young's inequality and Hardy-Littlewood-Sobolev inequality are proven. The Radon-Nikodym theorem, notions of harmonic analysis, classical inequalities and interpolation theorems, including Marcinkiewicz's theorem, the definition of Lebesgue points and Lebesgue differentiation theorem are further topics included. A detailed appendix provides the reader with various elements of elementary mathematics, such as a discussion around the calculation of antiderivatives or the Gamma function. The appendix also provides more advanced material such as some basic properties of cardinals and ordinals which are useful in the study of measurability.​

The Malliavin Calculus

Author: Denis R. Bell
Publisher: Courier Corporation
ISBN: 0486152057
Format: PDF, Kindle
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This introductory text presents detailed accounts of the different forms of the theory developed by Stroock and Bismut, discussions of the relationship between these two approaches, and a variety of applications. 1987 edition.

Differentiable Measures and the Malliavin Calculus

Author: Vladimir Igorevich Bogachev
Publisher: American Mathematical Soc.
ISBN: 082184993X
Format: PDF, Kindle
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This book provides the reader with the principal concepts and results related to differential properties of measures on infinite dimensional spaces. In the finite dimensional case such properties are described in terms of densities of measures with respect to Lebesgue measure. In the infinite dimensional case new phenomena arise. For the first time a detailed account is given of the theory of differentiable measures, initiated by S. V. Fomin in the 1960s; since then the method has found many various important applications. Differentiable properties are described for diverse concrete classes of measures arising in applications, for example, Gaussian, convex, stable, Gibbsian, and for distributions of random processes. Sobolev classes for measures on finite and infinite dimensional spaces are discussed in detail. Finally, we present the main ideas and results of the Malliavin calculus--a powerful method to study smoothness properties of the distributions of nonlinear functionals on infinite dimensional spaces with measures. The target readership includes mathematicians and physicists whose research is related to measures on infinite dimensional spaces, distributions of random processes, and differential equations in infinite dimensional spaces. The book includes an extensive bibliography on the subject.

The Regional Integration Manual

Author: Philippe De Lombaerde
Publisher: Routledge
ISBN: 1136702040
Format: PDF, ePub
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The Regional Integration Manual brings together different methods for monitoring and analysing regional integration processes in a systematic way. Employing a multi-disciplinary approach, it seeks to provide officials in regional organisations, researchers in think tanks, academics and students worldwide with an accessible set of both quantitative and qualitative tools, useful in their day-to-day work. The Manual addresses an increasing demand for such tools, in a world where mechanisms and ideas for effective regional government and governance are in dire need, whereas the monitoring and analytical capabilities of official and non-governmental actors often lag behind. It also addresses a rapidly growing academic community studying the determinants, depth, speed and other characteristics of regional integration and co-operation. Employing a multi-disciplinary approach, The Regional Integration Manual will be of interest to scholars of governance and regional politics as well as policy-makers and those in regional organisations.

Functions of Bounded Variation and Free Discontinuity Problems

Author: Luigi Ambrosio
Publisher: Courier Corporation
ISBN: 9780198502456
Format: PDF, ePub, Docs
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'... an excellent account of the theory of BV functions. It should serve as a standard reference, especially for the BV theory, for years to come.' Bulletin of the London Mathematical SocietyThis book deals with a class of mathematical problems which involve the minimization of the sum of a volume and a surface energy and have lately been refered to as 'free discontinuity problems'. Examples of such problems come from fracture mechanics, image analysis, or the theory of phase transitions. A systematic introduction to this field, this book is highly suitable for graduate students, bridging the gap between research level texts and elementary textbooks on measure theory and calculus of variation. The first half of the book contains a comprehensive and updated treatment of the theory of Functions of Bounded Variation and of the mathematical prerequisites of that theory, that is Abstract Measure Theory and Geometric Measure Theory.

Fully Nonlinear Elliptic Equations

Author: Luis A. Caffarelli
Publisher: American Mathematical Soc.
ISBN: 0821804375
Format: PDF, ePub, Docs
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The goal of the book is to extend classical regularity theorems for solutions of linear elliptic partial differential equations to the context of fully nonlinear elliptic equations. This class of equations often arises in control theory, optimization, and other applications. The authors give a detailed presentation of all the necessary techniques. Instead of treating these techniques in their greatest generality, they outline the key ideas and prove the results needed for developing the subsequent theory. Topics discussed in the book include the theory of viscosity solutions for nonlinear equations, the Alexandroff estimate and Krylov-Safonov Harnack-type inequality for viscosity solutions, uniqueness theory for viscosity solutions, Evans and Krylov regularity theory for convex fully nonlinear equations, and regularity theory for fully nonlinear equations with variable coefficients.

Submanifolds in Carnot Groups

Author: Davide Vittone
Publisher: Edizioni della Normale
ISBN:
Format: PDF, ePub, Mobi
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The book is devoted to the study of submanifolds in the setting of Carnot groups equipped with a sub-Riemannian structure; particular emphasis is given to the case of Heisenberg groups. A Geometric Measure Theory viewpoint is adopted, and features as intrinsic perimeters, Hausdorff measures, area formulae, calibrations and minimal surfaces are considered. Area formulae for the measure of submanifolds of arbitrary codimension are obtained in Carnot groups. Intrinsically regular hypersurfaces in the Heisenberg group are extensively studied: suitable notions of graphs are introduced, together with area formulae leading to the analysis of Plateau and Bernstein type problems.

Echoes of an Invisible World

Author: Jacomien Prins
Publisher: BRILL
ISBN: 9004281762
Format: PDF, Kindle
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In Echoes of an invisible world Jacomien Prins offers an account of the transformation of the notion of Pythagorean world harmony during the Renaissance and the role of the Italian philosophers Marsilio Ficino (1433-1499) and Francesco Patrizi (1529-1597) in redefining the relationship between cosmic order and music theory.