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Real Analysis: Measures, Integrals and Applications is devoted to the basics of integration theory and its related topics. The main emphasis is made on the properties of the Lebesgue integral and various applications both classical and those rarely covered in literature.   This book provides a detailed introduction to Lebesgue measure and integration as well as the classical results concerning integrals of multivariable functions. It examines the concept of the Hausdorff measure, the properties of the area on smooth and Lipschitz surfaces, the divergence formula, and Laplace's method for finding the asymptotic behavior of integrals. The general theory is then applied to harmonic analysis, geometry, and topology. Preliminaries are provided on probability theory, including the study of the Rademacher functions as a sequence of independent random variables.   The book contains more than 600 examples and exercises. The reader who has mastered the first third of the book will be able to study other areas of mathematics that use integration, such as probability theory, statistics, functional analysis, partial probability theory, statistics, functional analysis, partial differential equations and others.   Real Analysis: Measures, Integrals and Applications is intended for advanced undergraduate and graduate students in mathematics and physics. It assumes that the reader is familiar with basic linear algebra and differential calculus of functions of several variables.

Mathematical analysis --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- 517.1 Mathematical analysis --- Mathematics. --- Fourier analysis. --- Measure theory. --- Functions of real variables. --- Geometry. --- Measure and Integration. --- Fourier Analysis. --- Real Functions. --- Euclid's Elements --- Real variables --- Functions of complex variables --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Analysis, Fourier --- Math --- Science --- Mathematical analysis.

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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).

Measure theory --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Lebesgue measure --- Measurable sets --- Measure of a set --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Measure theory. --- Probabilities. --- Measure and Integration. --- Analysis. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Math --- Science --- 517.1 Mathematical analysis --- Mathematical analysis --- Global analysis (Mathematics). --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Probability Theory.

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The book begins at an undergraduate student level, assuming only basic knowledge of calculus in one variable. It rigorously treats topics such as multivariable differential calculus, the Lebesgue integral, vector calculus and differential equations. After having created a solid foundation of topology and linear algebra, the text later expands into more advanced topics such as complex analysis, differential forms, calculus of variations, differential geometry and even functional analysis. Overall, this text provides a unique and well-rounded introduction to the highly developed and multi-faceted subject of mathematical analysis as understood by mathematicians today.

Mathematical analysis. --- Mathematics. --- Matrix theory. --- Algebra. --- Functions of complex variables. --- Measure theory. --- Differential equations. --- Functions of real variables. --- Sequences (Mathematics). --- Real Functions. --- Linear and Multilinear Algebras, Matrix Theory. --- Measure and Integration. --- Functions of a Complex Variable. --- Ordinary Differential Equations. --- Sequences, Series, Summability. --- Mathematical sequences --- Numerical sequences --- Algebra --- Mathematics --- Real variables --- Functions of complex variables --- 517.91 Differential equations --- Differential equations --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Complex variables --- Elliptic functions --- Functions of real variables --- Mathematical analysis --- Math --- Science --- Differential Equations.

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Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings; that is, one-dimensional drums with fractal boundary. This second edition of Fractal Geometry, Complex Dimensions and Zeta Functions will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, complex analysis, distribution theory, and mathematical physics. The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. Key Features include: The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings · Complex dimensions of a fractal string are studied in detail, and used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal ·  Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula ·  The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field.

Fractals. --- Functions, Zeta. --- Geometry, Riemannian. --- Mathematics. --- Fractals --- Functions, Zeta --- Geometry, Riemannian --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Geometry --- Number theory. --- Zeta functions --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Number study --- Numbers, Theory of --- Riemann geometry --- Riemannian geometry --- Dynamics. --- Ergodic theory. --- Functional analysis. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Measure theory. --- Partial differential equations. --- Number Theory. --- Measure and Integration. --- Partial Differential Equations. --- Dynamical Systems and Ergodic Theory. --- Global Analysis and Analysis on Manifolds. --- Functional Analysis. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Partial differential equations --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Geometry, Differential --- Topology --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Math --- Science --- Dimension theory (Topology) --- Generalized spaces --- Geometry, Non-Euclidean --- Semi-Riemannian geometry --- Differential equations, partial. --- Differentiable dynamical systems. --- Global analysis. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics