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G1CMIN: MEASURE AND INTEGRATION 2002-3

URL: http://www.maths.nott.ac.uk/personal/jff/G1CMIN/index.html

Module Information
Handouts
Coursework
Assessment
Miscellaneous links of interest

Most of the printed handouts for this module are available on the web in two formats: ps (postscript) or pdf (portable document format). Which format is easiest to view will probably depend on the computer that you are using.

Module information for 2002 to 2003

Credits: 15
Duration: Approximately 33 lectures (including one revision session), three lectures a week in Spring Semester, starting Monday 27/1/2003
Lecturer: Dr J. F. Feinstein, room C301, email Joel.Feinstein@nottingham.ac.uk
Other staff involved: none.
Lecture times: (From week two onwards) Tuesday 9 in M&P Building room C12; Thursday 1 in M&P Building room C29 and Friday 12 in M&P Building room C29.
Provisional office hours (Spring Semester) Dr Feinstein expects to be in his office and available for consultation on Thursdays and Fridays usually between 10.00 AM and 12.00 noon. (See the web page http://www.maths.nott.ac.uk/personal/jff/ttnow.html for more details.)
Brief content description:
The module begins by introducing the concept of measure, a means for comparing the size of sets and generalizing intuitive notions such as length and area, and moves on to describe the elements of the Lebesgue theory of integration, which has wide generality, being applicable to a large class of functions, with `clean' properties which make it satisfying to work with. This will normally lead as far as the main convergence theorems. Lebesgue integration is a fundamental tool for advanced study in areas of mathematics such as functional analysis and potential theory, and provides the foundation for the axiomatic treatment of probability theory.
Prerequisites: G12RAN Real Analysis (or equivalent). I would advise that you should have obtained a reasonably good pass mark in G12RAN, for example about 55%, if you wish to attend the module G1CMIN.
Module aims: To teach the elements of measure theory and Lebesgue integration.
Module objectives: That the student should
- Be able to describe at least one approach to the construction of Lebesgue measure, the Lebesgue integral of a function and measure spaces.
- Know the principal theorems as treated and their proofs and be able to use them in the investigation of examples.
- Be able to prove simple unseen propositions concerning measure spaces, Lebesgue measure and integration.
Brief Syllabus:
- Chapter 1. The extended real line Formal manipulation of plus and minus infinity.
- Chapter 2. Classes of subsets of a set Semi-rings, rings and fields, sigma-fields etc.
- Chapter 3. Introduction to measures and measure spaces
- Chapter 4. The integral The Riemann integral revisited. Measurable functions. Integrals with respect to measures. The Lebesgue integral. Convergence theorems.
- Chapter 5. Outer measures and the construction of Lebesgue measure
Books: The following books are all worth looking at. The most appropriate books for this module are the book on Real and Complex Analysis by Rudin (highly recommended) and (for the theory of outer measures and the construction of Lebesgue measure) the book by Taylor.
1. R.B. Ash, Real Analysis and Probability, Academic Press.
2. C.W. Burrill, Measure, Integration and Probability, McGraw-Hill.
3. G. de Barra, Introduction to measure theory, van Nostrand.
4. P.R. Halmos, Measure Theory, van Nostrand.
5. W. Rudin, Principles of Mathematical Analysis, McGraw-Hill.
6. W. Rudin, Real and Complex Analysis, McGraw-Hill
7. S.J. Taylor, Introduction to Measure and Integration, CUP.

Handouts

Handouts will be issued in printed form, and will also be available (as they are issued) from the module web page.

Blow-by-blow account of the module, 2000-1 (including a list of key concepts and results): ps, pdf
This year's blow-by-blow account of the module (will be updated as the module progresses): ps, pdf
Module information sheet (this document).
Introduction and Chapter 1, The extended real line: ps, pdf
Chapter 2, Classes of sets: ps, pdf
Chapter 3, Measures and measure spaces: ps, pdf
Chapter 4, The Integral: ps, pdf
Approximation of f(x)=x by simple functions
Tutorial problems on Integration Theorems: ps, pdf
Extract from Chapter 5 on Lebesgue Measure: ps, pdf
Chapter 5, Outer measures and the construction of Lebesgue measure: ps, pdf
Tutorial problems on measures and outer measures: ps, pdf

Other handouts will be posted on the module web page as they appear.

Course work

Regular question sheets will be issued. Solutions should be handed in to me at the appointed time (shown on each sheet) for marking. The assignments form an essential part of the learning process but do not form part of your formal assessment. The question sheets will be available on the module web page as they are issued. The dates for handing in work will be: 7/2, 21/2, 7/3, 25/4 and 6/5 2003.

Question sheet 1: ps, pdf
Question sheet 2: ps, pdf
Question sheet 3: ps, pdf
Question sheet 4: ps, pdf
Question Sheet 5: ps, pdf

G1CMIN Assessment

Assessment will be by means of a two and a half hour examination in May/June. There will be five questions, and your best four answers will count. If you answer four of the questions perfectly, then you will obtain full marks. Please note that no calculators will be permitted in the examination. Should a resit examination be required in August/September it will take the same form as above (five questions, best four answers count, no calculators).
You may find it useful to look at the exam papers from previous years. Please note the format of the exam: three years ago there were six questions instead of five and you now have two and a half hours instead of two hours. Dr Feinstein gave this module in the years 2000-1 and 2001-2 and previously gave a similar module (G13AN4 Measure and Integration) in all of the years from 1992-3 to 1996-7 inclusive.

Supplementary Mathematics

For those taking this module as a 5 credit Supplementary Mathematics module, assessment will be by means of one special coursework assignment to be issued at the end of week 5 and to be handed in at the end of week 8. This will be based on the first three chapters of the module. Attendance at lectures after week 5 is optional for these students, as are the standard coursework assignments above.

Miscellaneous links which may be of interest:

G12RAN: Real Analysis
G13MTS: Metric and topological spaces [an error occurred while processing this directive]

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