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Materials available from this web page are copyright © J.F. Feinstein unless otherwise stated.
The lecture notes/module slides will be available here, as will the annotated slides from classes. The slides have a very large font, which may be useful to students with imperfect eyesight. However, you may wish to consider printing multiple pages per sheet.
There will also be podcasts (audio) of at least some of the classes.| Date (and link to mp3 file) | Slides covered |
|---|---|
| 21/1/08 (mp3 file) | Introductory lecture: written slides |
| 25/1/08 a (mp3 file) | Functions, sets, countability and uncountability, Slides 1-6: 07-08 annotated slides |
| 25/1/08 b (mp3 file) | Revision session on mathematical analysis from earlier modules: written slides |
| 28/1/08 (mp3 file) | Functions, sets, countability and uncountability, Slides 6-10: 07-08 annotated slides |
| 1/2/08 a (mp3 file) | Functions, sets, countability and uncountability, Slides 10-12: 07-08 annotated slides |
| 1/2/08 b (mp3 file) | Functions, sets, countability and uncountability, Conclusion: 07-08 annotated slides |
| 1/2/08 c (mp3 file) | Section 1, The extended real line, Slides 1-4: 07-08 annotated slides |
| 4/2/08 (mp3 file) | Section 1, The extended real line, Slides 4-9: 07-08 annotated slides |
| 8/2/08 a (mp3 file) | Discussion of Question Sheet 2: written slides |
| 8/2/08 b (mp3 file) | Section 1, The extended real line,
Slides 9-11: 07-08
annotated slides Remaining audio for Chapter 1 slides is available as part of the Lecture 7 audio podcast from 2006-7 |
| 11/2/08 (mp3 file) | Discussion of materials from 2006-7, up to and including Lecture 8: written slides |
| 15/2/08 a (mp3 file) | Discussion of Question Sheet 3, inf and sup: written slides |
| 15/2/08 b (mp3 file) | Examples and non-examples of semi-rings, rings and fields: written slides |
| 18/2/08 (mp3 file) | Semi-rings, rings, fields and sigma-fields, especially the Borel sets: written slides |
| 22/2/08 (mp3 file) | Discussion of the properties of image and pre-image: written slides |
| 25/2/08 (mp3 file) | Discussion of Lebesgue outer measure, series with all terms equal, counting measure, and Sheet 4, Question 2: written slides |
| 29/2/08 (mp3 file) | Examples Class on Sheet 4, Questions 4 and 6: written slides |
| 29/2/08 b (mp3 file) | Further discussion of sigma-fields generated by collections of sets, and a proof that counting measure really is a measure (case by case analysis): written slides |
| 3/3/08 (mp3 file) | Properties that hold almost everywhere. Sigma-finiteness or otherwise of counting measure. Failure of Riemann integrability for the characteristic function of the rationals: written slides |
| 7/3/08 a (mp3 file) | Examples Class on Sheet 5, covering Questions 1 and 3: written slides |
| 7/3/08 b (mp3 file) | Measurable functions and non-measurable functions, including Sheet 5, Question 5: written slides |
| 10/3/08 (mp3 file) | Monotone approximation from below by simple
functions: special cases of f(x) = x and f(x)=x
squared.. Discussion of functions which take, instead, countably many different values: written slides |
| 14/3/08 a (mp3 file) | Examples Class on Sheet 6, covering Questions 4 and 6: annotated slides |
| 14/3/08 b (mp3 file) | Continuity properties of measures: Proposition 3.6(i) for sigma-fields: written slides |
| 14/4/08 (mp3 file) | Old and new notation for the integral (especially non-negative, simple measurable functions). Examples where the integral of the limit is not equal to the limit of the integrals: written slides |
| 18/4/08 a (mp3 file) | Worksheet on Fatou's Lemma: calculations involving Lebesgue measure and Lebesgue integrals of specific sets/functions. Examples where the inequality in Fatou's Lemma is strict: annotated slides |
| 18/4/08 b (mp3 file) | Questions and solutions building on definitions of measurable sets and functions, and properties which hold almost everywhere. Non-negative measurable functions whose integral is finite must be finite almost everywhere: written slides |
| 21/4/08 (mp3 file) | Discussion of positive and negative parts of functions. Discussion of 4.36: the modulus of the integral is at most equal to the integral of the modulus. Discussion of L1, and why there is no real loss in insisting that L1 functions are real-valued rather than extended real-valued: written slides |
| 25/4/08 a (mp3 file) | Examples Class focussing on Question sheet 7, Questions 3, 4 and 5: written slides |
| 25/4/08 b (mp3 file) | Discussion of the relationships between (proper) Riemann integrals, Lebesgue integrals and improper Riemann integrals: written slides |
| 28/4/08 (mp3 file) | Examples Class on measures, outer measures, monotonicity, finite additivity and countable subadditivity: annotated slides |
| 2/5/08 a (mp3 file) | Examples concerning Lebesgue outer measure, working directly from
the definition. Discussion of measurability (Proposition 4.8): written slides |
| 2/5/08 b (mp3 file) | Discussion of the relationships between the various types of integrals: A-level, G11CAL, Riemann, Lebesgue. Example concerning outer measures and measurable sets: written slides |
| 9/5/08 (mp3 file) | (11AM only) Optional revision/Q&A session: written slides |
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