LECTURE NOTES OF WILLIAM CHEN
INTRODUCTION TO LEBESGUE INTEGRATION
This set of notes was mainly written in 1977 while the author was an undergraduate at Imperial College, University of London. Chapters 1 and 3 were first used in lectures given there in 1982 and 1983, while Chapter 2 was added in Sydney in 1996.
The material has been organized in such a way to create a single volume suitable for an introduction to some of the basic ideas in Lebesgue integration with the minimal use of measure theory.
To read the notes, click the links below for connection to the appropriate PDF files.
The material is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, the documents may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners.
Chapter 1 : THE REAL NUMBERS AND COUNTABILITY >>
- Introduction
- Completeness of the Real Numbers
- Consequences of the Completeness Axiom
- Countability
Chapter 2 : THE RIEMANN INTEGRAL >>
- Riemann Sums
- Lower and Upper Integrals
- Riemann Integrability
- Further Properties of the Riemann Integral
- An Important Example
Chapter 3 : POINT SETS >>
- Open and Closed Sets
- Sets of Measure Zero
- Compact Sets
Chapter 4 : THE LEBESGUE INTEGRAL >>
- Step Functions on an Interval
- Upper Functions on an Interval
- Lebesgue Integrable Functions on an Interval
- Sets of Measure Zero
- Relationship with Riemann Integration
Chapter 5 : MONOTONE CONVERGENCE THEOREM >>
- Step Functions on an Interval
- Upper Functions on an Interval
- Lebesgue Integrable Functions on an Interval
Chapter 6 : DOMINATED CONVERGENCE THEOREM >>
- Lebesgue's Theorem
- Consequences of Lebesgue's Theorem
Chapter 7 : LEBESGUE INTEGRALS ON UNBOUNDED INTERVALS >>
- Some Limiting Cases
- Improper Riemann Integrals
Chapter 8 : MEASURABLE FUNCTIONS AND MEASURABLE SETS >>
- Measurable Functions
- Further Properties of Measurable Functions
- Measurable Sets
- Additivity of Measure
- Lebesgue Integrals over Measurable Sets
Chapter 9 : CONTINUITY AND DIFFERENTIABILITY OF LEBESGUE INTEGRALS >>
- Continuity
- Differentiability
Chapter 10 : DOUBLE LEBESGUE INTEGRALS >>
- Introduction
- Decomposition into Squares
- Fubini's Theorem for Step Functions
- Sets of Measure Zero
- Fubini's Theorem for Lebesgue Integrable Functions