LECTURE NOTES OF WILLIAM CHEN
# INTRODUCTION TO LEBESGUE INTEGRATION

### Chapter 1 : THE REAL NUMBERS AND COUNTABILITY >>

### Chapter 2 : THE RIEMANN INTEGRAL >>

### Chapter 3 : POINT SETS >>

### Chapter 4 : THE LEBESGUE INTEGRAL >>

### Chapter 5 : MONOTONE CONVERGENCE THEOREM >>

### Chapter 6 : DOMINATED CONVERGENCE THEOREM >>

### Chapter 7 : LEBESGUE INTEGRALS ON UNBOUNDED INTERVALS >>

### Chapter 8 : MEASURABLE FUNCTIONS AND MEASURABLE SETS >>

### Chapter 9 : CONTINUITY AND DIFFERENTIABILITY OF LEBESGUE INTEGRALS >>

### Chapter 10 : DOUBLE LEBESGUE INTEGRALS >>

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.

- Introduction
- Completeness of the Real Numbers
- Consequences of the Completeness Axiom
- Countability

- Riemann Sums
- Lower and Upper Integrals
- Riemann Integrability
- Further Properties of the Riemann Integral
- An Important Example

- Open and Closed Sets
- Sets of Measure Zero
- Compact Sets

- Step Functions on an Interval
- Upper Functions on an Interval
- Lebesgue Integrable Functions on an Interval
- Sets of Measure Zero
- Relationship with Riemann Integration

- Step Functions on an Interval
- Upper Functions on an Interval
- Lebesgue Integrable Functions on an Interval

- Lebesgue's Theorem
- Consequences of Lebesgue's Theorem

- Some Limiting Cases
- Improper Riemann Integrals

- Measurable Functions
- Further Properties of Measurable Functions
- Measurable Sets
- Additivity of Measure
- Lebesgue Integrals over Measurable Sets

- Continuity
- Differentiability

- Introduction
- Decomposition into Squares
- Fubini's Theorem for Step Functions
- Sets of Measure Zero
- Fubini's Theorem for Lebesgue Integrable Functions