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This book provides an introduction to Lebesgue measure and integration as well as the classical results concerning integrals of multivariable functions. It features over 600 examples.

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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.

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Measure.- The Lebesgue Model.- Measurable Functions.- The Integral.- The Product Measure.- Change of Variables in an Integral.- Integrals Dependent on a Parameter.- Surface Integrals.- Approximation and Convolution of the Space.- Fourier Series and the Fourier Transform.- Charges. The Radon-Nikodym Theory.- Integral Representation of Linear Functionals.- Appendices.

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"Written in a didactic style, with clear proofs and intuitive motivations for the abstract notions, the book is a valuable addition to the literature on measure theory and integration and their applications to various areas of analysis and geometry. The numerous nontrivial examples and applications are of great importance for those interested in various domains of modern analysis and geometry, or in teaching." (S. Cobzas, Studia Universitatis Babes-Bolyia, Mathematica, Vol. 60 (1), 2015)
"The book contains enough material for a good three-semester graduate course in analysis. Complete proofs are given for all results, and the reader-friendly, exposition style presents lots of details and motivational tips throughout. ... Summing Up: Highly recommended. Graduate students." (D. M. Ha, Choice, Vol. 51 (10), June, 2014)

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