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G1CMIN: MEASURE AND INTEGRATION 2002-3
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
Duration: Approximately 33 lectures (including one revision session),
three lectures a week in
Spring Semester, starting Monday 27/1/2003
Dr J. F. Feinstein, room C301, email
Other staff involved: none.
(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
Thursdays and Fridays
usually between 10.00 AM and 12.00 noon. (See the web page
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
G12RAN Real Analysis
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
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.
- 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.
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.
- R.B. Ash, Real Analysis and Probability, Academic Press.
- C.W. Burrill, Measure, Integration and Probability, McGraw-Hill.
- G. de Barra, Introduction to measure theory, van Nostrand.
- P.R. Halmos, Measure Theory, van Nostrand.
- W. Rudin, Principles of Mathematical Analysis, McGraw-Hill.
- W. Rudin, Real and Complex Analysis, McGraw-Hill
- S.J. Taylor, Introduction to Measure and Integration, CUP.
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):
- This year's blow-by-blow account of the module (will be updated
as the module progresses):
Module information sheet (this document).
- Introduction and Chapter 1, The extended real line:
- Chapter 2, Classes of sets:
- Chapter 3, Measures and measure spaces:
- Chapter 4, The Integral:
- Approximation of f(x)=x by simple functions
- Tutorial problems on Integration Theorems:
- Extract from Chapter 5 on Lebesgue Measure:
- Chapter 5, Outer measures and the construction of Lebesgue measure:
- Tutorial problems on measures and outer measures:
Other handouts will be posted on the module web page as they appear.
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:
- Question sheet 2:
- Question sheet 3:
- Question sheet 4:
- Question Sheet 5:
Assessment will be by means of a two and a half hour examination in
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
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.
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:
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