The material covered in these notes is essentially the development of the theory of measure and integration; the lack of time means that there is little in the way of applications of the theory and thus little motivation for the student. I would hope that the material presented in the appendices will show the wide range of applications of the theory.
| Notes | Contents |
|---|---|
| Notes 1 | 1 Classes of subsets: 1.1 Topology, 1.2 Rings, 1.3 Fields. (8 pages) |
| Appendix 1 | Cardinality, Topological Space results. (6 pages.) |
| Notes 2 | 2 Set functions: 2.1 (finitely) additive functions, 2.2 sigma-additive functions, 2.3 Extending a sigma-additive function, 2.4 Measure and Outer measure. (10 pages) |
| Notes 3a Notes 3b | 2.5 Outer measure and Measurable sets, 2.6 Lebesgue Measurable sets, 2.7 Non-measurable sets, 2.8 Sets of measure zero. (6 and 6 pages) |
| Notes 4 | 3 Measurable functions: 3.1 Sequences of functions. (11 pages) |
| Notes 5 | 3.2 Simple functions. (4 pages) |
| Appendix 5 | An interesting simple function. (2 pages.) |
| Notes 6 | 4 Integration: 4.1 Integration of non-negative simple functions. (9 pages) |
| Appendix 6 | Chebychev's Theorem and an application to Normal Numbers. (5 pages.) |
| Notes 7 | 4.3 Interchanging integrals with other operations. (8 pages) |
| Appendix 7 | Extended version of Monotonic Convergence Theorem with yet another proof that ∑(1/n2)=π2/6. (7 pages.) |
| Notes 8 | 4.4 Integration of measurable functions. (8 pages) |
| Appendix 8a | Comparison of the Riemann and Lebesgue integrals, Measure Preserving Transformations including Poincare's Recurrence Theorem. (10 pages.) |
| Appendix 8b | Spaces of integrable functions, showing that these spaces are complete. (7 pages.) |
| Notes 9 | 5 Product Spaces and Fubini's Theorem. |
| Appendix 9 | Completion of product spaces. (3 pages.) |
There is one further appendix dealing with applications to number theory and in particular rational approximations to irrationals. (8 pages.)
There is one file with all the questions for the course. (9 pages.)
The solutions are split into three files.