Introduction to Lebesgue Integration

by WWL Chen

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 chapters below for connection to the appropriate PDF files. You will need Adobe Acrobat Reader Version 4.0 or later.

The material is available free to all individuals, on the understanding that it is not to be used for financial gains, 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 (last uploaded on 2 December 2002)

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

Chapter 2: THE RIEMANN INTEGRAL (last uploaded on 2 December 2002)

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

Chapter 3: POINT SETS (last uploaded on 2 December 2002)

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

Chapter 4: THE LEBESGUE INTEGRAL (last uploaded on 2 December 2002)

• 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 (last uploaded on 2 December 2002)

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

Chapter 6: DOMINATED CONVERGENCE THEOREM (last uploaded on 2 December 2002)

• Lebesgue's Theorem
• Consequences of Lebesgue's Theorem

Chapter 7: LEBESGUE INTEGRALS ON UNBOUNDED INTERVALS (last uploaded on 2 December 2002)

• Some Limiting Cases
• Improper Riemann Integrals

Chapter 8: MEASURABLE FUNCTIONS AND MEASURABLE SETS (last uploaded on 2 December 2002)

• 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 (last uploaded on 2 December 2002)

• Continuity
• Differentiability

Chapter 10: DOUBLE LEBESGUE INTEGRALS (last uploaded on 2 December 2002)

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