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The purpose is to present a complete course on global analysis topics and establish some orbital applications of the integration on topological groups and their algebras to harmonic analysis and induced representations in representation theory.

Mathematics. --- Math --- Science --- Integral Calculus --- Physical Sciences --- Engineering and Technology --- Mathematics --- Analysis & Calculus

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Integrals --- Data processing. --- -517.3 --- 681.3*I10 --- Calculus, Integral --- Data processing --- Integral calculus. Integration --- Computerwetenschap--?*I10 --- 517.3 Integral calculus. Integration --- 517.3 --- Integrals - Data processing. --- Integrals-Data processing

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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.

Integral calculus & equations --- Measure theory. --- Measure and Integration. --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Mathematics --- Measure theory

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Differential equations --- Integral equations --- #KVIV --- 517.3 --- 517.9 --- Equations, Integral --- Functional equations --- Functional analysis --- Integral calculus. Integration --- Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Integral equations. --- 517.9 Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- 517.3 Integral calculus. Integration --- Equations integrales

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517.3 --- Integral calculus. Integration --- Harmonic analysis. --- Integrals. --- 517.3 Integral calculus. Integration --- Harmonic analysis --- Integrals --- Calculus, Integral --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Differential equations --- Intégrales singulières