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About this book :-
Introduction to Measure Theory and Integration written by
Luigi Ambrosio .
Book Detail :-
Title: Introduction to Measure Theory and Integration
Edition:
Author(s): Luigi Ambrosio
Publisher:
Series:
Year:
Pages:
Type: PDF
Language: English
ISBN:
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Book Contents :-
Introduction to Measure Theory and Integration written by
Luigi Ambrosio
cover the following topics.
Preface
Introduction
Measure spaces
Notation and preliminaries, Rings, algebras and σ–algebras, Additive and σ–additive functions, Measurable spaces and measure spaces, The basic extension theorem, Dynkin systems, The outer measure, The Lebesgue measure in R, Inner and outer regularity of measures on metric spaces
Integration
Inverse image of a function, Measurable and Borel functions, Partitions and simple functions, Integral of a nonnegative E –measurable function, Integral of simple functions, The repartition function, The archimedean integral, Integral of a nonnegative measurable function, Integral of functions with a variable sign, Convergence of integrals, Uniform integrability and Vitali convergence theorem, A characterization of Riemann integrable functions
Spaces of integrable functions
Spaces L p(X, E, μ) and L p(X, E, μ), The Lp norm, Holder and Minkowski inequalities, Convergence in L p(X, E, μ) and completeness, The space L∞(X, E , μ), Dense subsets of L p(X, E , μ)
Hilbert spaces
Scalar products, pre-Hilbert and Hilbert spaces, The projection theorem, Linear continuous functionals, Bessel inequality, Parseval identity and orthonormal systems, Hilbert spaces on C
Fourier series
Pointwise convergence of the Fourier series, Completeness of the trigonometric system, Uniform convergence of the Fourier series
Operations on measures
The product measure and Fubini–Tonelli theorem, The Lebesgue measure on Rn, Countable products, Comparison of measures, Signed measures, Measures in R, Convergence of measures on R, Fourier transform, Fourier transform of a measure
The fundamental theorem of the integral calculus
Measurable transformations
Image measure, Change of variables in multiple integrals, Image measure of L n by a C1 diffeomorphism
Continuity and differentiability of functions depending on a parameter
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