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2018 | Book

Measures Read chapter Read first chapter

Lecture Notes in Real Analysis

Author: Xiaochang Wang

Publisher: Springer International Publishing

Book Series : Compact Textbooks in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This compact textbook is a collection of the author’s lecture notes for a two-semester graduate-level real analysis course. While the material covered is standard, the author’s approach is unique in that it combines elements from both Royden’s and Folland’s classic texts to provide a more concise and intuitive presentation. Illustrations, examples, and exercises are included that present Lebesgue integrals, measure theory, and topological spaces in an original and more accessible way, making difficult concepts easier for students to understand. This text can be used as a supplementary resource or for individual study.

Table of Contents

Frontmatter
Chapter 1. Measures
Abstract
To understand why we need a theory of measures and a new way to define integrals (Lebesgue integrals), we need to look what is wrong with Riemann’s theory of integrals.
Xiaochang Wang
Chapter 2. Integration
Abstract
Recall that in order to define the Lebesgue integral for a function f, the set
Xiaochang Wang
Chapter 3. Signed Measures and Differentiation
Abstract
Let \((X,{\mathcal M})\) be a measurable space. A signed measure of \((X,{\mathcal M})\) is a countably additive set function \(\nu :{\mathcal M}\to [-\infty ,\infty )\) or (−∞, ∞] such that ν(∅) = 0.
Xiaochang Wang
Chapter 4. Topology: A Generalization of Open Sets
Abstract
A collection \({\mathcal T}\) of subsets of X is called a topology on X if
Xiaochang Wang
Chapter 5. Elements of Functional Analysis
Abstract
Let X be a vector space over \({\mathbb R} \). A norm ∥⋅∥ on X is a function X → [0, ∞) such that
Xiaochang Wang
Chapter 6. Lp Spaces
Abstract
Let \((X,{\mathcal M},\mu )\) be a measure space and p > 0.
Xiaochang Wang
Backmatter
Metadata
Title
Lecture Notes in Real Analysis
Author
Xiaochang Wang
Copyright Year
2018
Electronic ISBN
978-3-319-98956-3
Print ISBN
978-3-319-98955-6
DOI
https://doi.org/10.1007/978-3-319-98956-3

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