MA40042 Measure Theory and Integration

This page is for the module MA40042 Measure Theory and Integration as taught at the University of Bath in 2016–17. I have no idea what relation it bears to any currently taught material.

The module is split into two parts:

• Measure theory is the first part of the course, is lectured by me, and lasts five weeks;
• Integration is the second part of the course, will be lectured by Dr Matt Roberts, and will last six weeks.

This page is for the measure theory part of the course only.

Lecture notes

• Lecture 0: Introduction (Lecture notes, Handout: Conventions regarding infinity)
• Lecture 1: σ-algebras (Lecture notes)
• Lecture 2: The Borel σ-algebra (Lecture notes)
• Lecture 3: Measures (Lecture notes)
• Lecture 4: Constructing measures I: Outer measure (Lecture notes)
• Lecture 5: Constructing measures II: Measurable sets (Lecture notes, Handout: Carathéodory’s splitting condition)
• Lecture 6: Constructing measures III: Carathéodory’s extension theorem (Lecture notes, Handout: Constructing measures: an outline)
• Lecture 7: Proof of Carathéodory’s extension theorem (Lecture notes)
• Lecture 8: π-systems, λ-systems, and the uniqueness lemma (Lecture notes, Handout: Set systems)
• Lecture 9: More on the Lebesgue measure

Problem sheets

There is one problem sheet per week.

• Problem Sheet 1
• Problem Sheet 2 – Solutions to Question 3
• Problem Sheet 3 – Solutions to Questions 2 and 3
• Problem Sheet 4 – Solutions to Questions 3
• Problem Sheet 5 (This problem sheet has been lost to the mists of time…)

Past exam questions for the course are available online.

Books

The lectures should cover all necessary material for the course. However, it may be helpful to consult a textbook for further information or a different view of the material.

I have found the following books useful:

• D Williams, Probability with Martingales, Cambridge University Press, 1991. [Chapters 1 and A1]
• T Lindstrøm, Mathematical Analysis. Available online [Chapters 5 and 6]

I really liked Lindstrøm, and you will notice many similarities to the lecture notes. Williams takes a more probabilistic view of material, which some may prefer.

I haven’t used them much myself, but hear good things about:

• P Billingsley, Probability and Measure, anniversary edition, Wiley, 2012. [Chapters 1 and 2]
• R Durrett, Probability: Theory and examples, fourth edition, Cambridge University Press, 2010. [Chapter 1 and Appendix A]
• W Rudin, Real and Complex Analysis, third edition, McGraw-Hill, 1987. [Chapters 1 and 2]

All these books cover relevant material for the Integration part of the course also.