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Introduction to Measure Theory and Integration / by Luigi Ambrosio, Giuseppe Prato, Andrea Mennucci
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Appunti/Lecture Notes ; 10 | SpringerLink BücherPublisher: Pisa : Edizioni della Normale, 2011Description: Online-Ressource (198p, digital)ISBN:- 9788876423864
- 510
- 515.42
- QA312-312.5
- QA312 .A42 2011
Contents:
Summary: Measure spaces -- Integration -- Spaces of measurable functions -- Hilbert spaces -- Fourier series -- Operations on measures -- The fundamental theorem of integral calculus -- Operations on measures -- Appendix: Riesz representation theorem of the dual of C(K) and integrals depending on a parameter -- Solutions to the exercises.Summary: 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.PPN: PPN: 1651332215Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Title Page; Copyright Page; Table of Contents; Preface; Introduction; Chapter 1 Measure spaces; 1.1. Notation and preliminaries; 1.2. Rings, algebras and s-algebras; 1.3. Additive and s-additive functions; 1.4. Measurable spaces and measure spaces; 1.5. The basic extension theorem; 1.5.1. Dynkin systems; 1.5.2. The outer measure; 1.6. The Lebesgue measure in R; 1.7. Inner and outer regularity of measures on metric spaces; Chapter 2 Integration; 2.1. Inverse image of a function; 2.2. Measurable and Borel functions; 2.3. Partitions and simple functions
2.4. Integral of a nonnegative E -measurable function2.4.1. Integral of simple functions; 2.4.2. The repartition function; 2.4.3. The archimedean integral; 2.4.4. Integral of a nonnegative measurable function; 2.5. Integral of functions with a variable sign; 2.6. Convergence of integrals; 2.6.1. Uniform integrability and Vitali convergence theorem; 2.7. A characterization of Riemann integrable functions; Chapter 3 Spaces of integrable functions; 3.1. Spaces L p(X, E,µ) and L p(X, E,µ); 3.2. The L p norm; 3.2.1. Hölder and Minkowski inequalities
3.3. Convergence in L p(X,E ,µ) and completeness3.4. The space L8(X,E ,µ); 3.5. Dense subsets of L p(X, E,µ); Chapter 4 Hilbert spaces; 4.1. Scalar products, pre-Hilbert and Hilbert spaces; 4.2. The projection theorem; 4.3. Linear continuous functionals; 4.4. Bessel inequality, Parseval identity and orthonormal systems; 4.5. Hilbert spaces on C; Chapter 5 Fourier series; 5.1. Pointwise convergence of the Fourier series; 5.2. Completeness of the trigonometric system; 5.3. Uniform convergence of the Fourier series; Chapter 6 Operations on measures
6.1. The product measure and Fubini-Tonelli theorem6.2. The Lebesgue measure on Rn; 6.3. Countable products; 6.4. Comparison of measures; 6.5. Signed measures; 6.6. Measures in R; 6.7. Convergence of measures on R; 6.8. Fourier transform; 6.8.1. Fourier transform of a measure; Chapter 7 The fundamental theorem of the integral calculus; Chapter 8 Measurable transformations; 8.1. Image measure; 8.2. Change of variables in multiple integrals; 8.3. Image measure of L n by a C1 diffeomorphism; Appendix A; A.1. Continuity and differentiability of functions depending on a parameter
A.2. The dual space of continuous functionsSolutions of some exercises; References; LECTURE NOTES;
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