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The Lebesgue integral is now standard for both applications and advanced mathematics. This books starts with a review of the familiar calculus integral and then constructs the Lebesgue integral from the ground up using the same ideas. A Primer of Lebesgue Integration has been used successfully both in the classroom and for individual study.Bear presents a clear and simple introduction for those intent on further study in higher mathematics. Additionally, this book serves as a refresher providing new insight for those in the field. The author writes with an engaging, commonsense style t

Lebesgue integral. --- Integration, Lebesgue --- L-integral --- Lebesgue integration --- Lebesgue-Stieltjes integral --- Lebesgue's integral --- Stieltjes integral, Lebesgue --- -Lebesgue integral. --- -Integrals, Generalized --- Measure theory

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Operator theory --- Measure theory. Mathematical integration --- Lebesgue integral --- Radon measures --- Banach lattices --- Lebesgue, Intégrale de. --- Radon, Mesures de. --- Banach, Treillis de. --- Measures, Radon --- Measure theory --- Vector-valued measures --- Integration, Lebesgue --- L-integral --- Lebesgue integration --- Lebesgue-Stieltjes integral --- Lebesgue's integral --- Stieltjes integral, Lebesgue --- -Integrals, Generalized --- Lattices, Banach --- Banach algebras --- Lattice theory

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The main topics of this book are convergence and topologization. Integration on a compact interval on the real line is treated with Riemannian sums for various integration bases. General results are specified to a spectrum of integrations, including Lebesgue integration, the Denjoy integration in the restricted sense, the integrations introduced by Pfeffer and by Bongiorno, and many others. Morever, some relations between integration and differentiation are made clear.The book is self-contained. It is of interest to specialists in the field of real functions, and it can also be read by student

Lebesgue integral. --- Henstock-Kurzweil integral. --- Vector spaces. --- Linear spaces --- Linear vector spaces --- Algebras, Linear --- Functional analysis --- Vector analysis --- Gauge integral --- Generalized Riemann integral --- Henstock integrals --- HK integral --- Kurzweil-Henstock integral --- Kurzweil integral --- Riemann integral, Generalized --- Integrals, Generalized --- Integration, Lebesgue --- L-integral --- Lebesgue integration --- Lebesgue-Stieltjes integral --- Lebesgue's integral --- Stieltjes integral, Lebesgue --- -Integrals, Generalized --- Measure theory --- Henstock-Kurzweil integral --- Vector spaces

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This textbook, based on three series of lectures held by the author at the University of Strasbourg, presents functional analysis in a non-traditional way by generalizing elementary theorems of plane geometry to spaces of arbitrary dimension. This approach leads naturally to the basic notions and theorems. Most results are illustrated by the small ℓp spaces. The Lebesgue integral, meanwhile, is treated via the direct approach of Frigyes Riesz, whose constructive definition of measurable functions leads to optimal, clear-cut versions of the classical theorems of Fubini-Tonelli and Radon-Nikodým. Lectures on Functional Analysis and the Lebesgue Integral presents the most important topics for students, with short, elegant proofs. The exposition style follows the Hungarian mathematical tradition of Paul Erdős and others. The order of the first two parts, functional analysis and the Lebesgue integral, may be reversed. In the third and final part they are combined to study various spaces of continuous and integrable functions. Several beautiful, but almost forgotten, classical theorems are also included. Both undergraduate and graduate students in pure and applied mathematics, physics and engineering will find this textbook useful. Only basic topological notions and results are used and various simple but pertinent examples and exercises illustrate the usefulness and optimality of most theorems. Many of these examples are new or difficult to localize in the literature, and the original sources of most notions and results are indicated to help the reader understand the genesis and development of the field.

Mathematics. --- Approximation theory. --- Functional analysis. --- Measure theory. --- Functional Analysis. --- Measure and Integration. --- Approximations and Expansions. --- Lebesgue measure --- Measurable sets --- Measure of a set --- Functional calculus --- Theory of approximation --- Math --- Calculus of variations --- Functional equations --- Integral equations --- Science --- Lebesgue integral. --- Integration, Lebesgue --- L-integral --- Lebesgue integration --- Lebesgue-Stieltjes integral --- Lebesgue's integral --- Stieltjes integral, Lebesgue --- -Integrals, Generalized --- Measure theory --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra)

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This book provides an accessible introduction to the theory of variable Lebesgue spaces. These spaces generalize the classical Lebesgue spaces by replacing the constant exponent p with a variable exponent p(x). They were introduced in the early 1930s but have become the focus of renewed interest since the early 1990s because of their connection with the calculus of variations and partial differential equations with nonstandard growth conditions, and for their applications to problems in physics and image processing. The book begins with the development of the basic function space properties. It avoids a more abstract, functional analysis approach, instead emphasizing an hands-on approach that makes clear the similarities and differences between the variable and classical Lebesgue spaces. The subsequent chapters are devoted to harmonic analysis on variable Lebesgue spaces. The theory of the Hardy-Littlewood maximal operator is completely developed, and the connections between variable Lebesgue spaces and the weighted norm inequalities are introduced. The other important operators in harmonic analysis - singular integrals, Riesz potentials, and approximate identities - are treated using a powerful generalization of the Rubio de Francia theory of extrapolation from the theory of weighted norm inequalities. The final chapter applies the results from previous chapters to prove basic results about variable Sobolev spaces.

Lebesgue integral. --- Harmonic analysis. --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Integration, Lebesgue --- L-integral --- Lebesgue integration --- Lebesgue-Stieltjes integral --- Lebesgue's integral --- Stieltjes integral, Lebesgue --- -Integrals, Generalized --- Measure theory --- Functional analysis. --- Global analysis. --- Abstract Harmonic Analysis. --- Functional Analysis. --- Global Analysis and Analysis on Manifolds. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic