Lebesgue Integral (Compact Textbooks in Mathematics)

✍ Scribed by Liviu C. Florescu

• Publisher: Birkhäuser
• Year: 2021
• Tongue: English
• Leaves: 224
• Category: Library

✦ Synopsis

This book presents a compact and self-contained introduction to the theory of measure and integration. The introduction into this theory is as necessary (because of its multiple applications) as difficult for the uninitiated. Most measure theory treaties involve a large amount of prerequisites and present crucial theoretical challenges. By taking on another approach, this textbook provides less experienced readers with material that allows an easy access to the definition and main properties of the Lebesgue integral.

The book will be welcomed by upper undergraduate/early graduate students who wish to better understand certain concepts and results of probability theory, statistics, economic equilibrium theory, game theory, etc., where the Lebesgue integral makes its presence felt throughout. The book can also be useful to students in the faculties of mathematics, physics, computer science, engineering, life sciences, as an introduction to a more in-depth study of measure theory.

✦ Table of Contents

Introduction
Contents
List of Figures
1 Lebesgue Measure on R
1.1 Measure of Open Sets
1.2 Lebesgue Outer Measure
1.3 Lebesgue Measurable Sets
1.4 Abstract Setting
1.5 Exercises
2 Measurable Functions
2.1 Definitions. Properties
2.1.1 Operations with Measurable Functions
2.2 Different Types of Convergence
2.3 The Structure of Measurable Functions
2.4 Abstract Setting
2.5 Exercises
3 Lebesgue Integral
3.1 Integrals of Nonnegative Measurable Functions
3.2 Integrable Functions. Lebesgue Integral
3.3 The Space of Integrable Functions
3.4 Comparison with the Riemann Integral
3.5 Properties of the Lebesgue Integral
3.5.1 Change of Variables
3.5.2 Integrals Depending on a Parameter
3.5.3 Jensen's Inequality
3.6 Abstract Setting
3.7 Exercises
4 The Lp Spaces
4.1 Algebraic and Topological Structure
4.2 Density Properties in Lp
4.3 The L∞ Space
4.4 Fourier Series in L2([-π,π])
4.5 Abstract Setting
4.6 Exercises
5 Lebesgue Integral on R2
5.1 Lebesgue Measure on R2
5.2 Lebesgue Multiple Integrals
5.3 Fubini's Theorem
5.4 Abstract Setting
5.5 Exercises
6 Signed Measures
6.1 Decomposition Theorems
6.2 Radon-Nikodym Theorem
6.3 The Integral and the Derivative
6.4 Exercises
7 Appendices
7.1 Riemann Integral
7.2 Pompeiu's Function
7.3 Differentiability of Monotonic Functions
Bibliography
Index