| Unit name | Analysis 3 |
|---|---|
| Unit code | MATH34000 |
| Credit points | 20 |
| Level of study | H/6 |
| Teaching block(s) |
Teaching Block 1 (weeks 1 - 12) |
| Unit director | Professor. van den Berg |
| Open unit status | Not open |
| Pre-requisites |
Level 2 Analysis and Linear Algebra |
| Co-requisites |
None |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
This course introduces the Lebesgue integral and develops the elements of measure theory. Topics will include measurable functions, measures, Lebesgue integral, monotone convergence theorem, Fatou's lemma, Lebesgue dominated convergence, Lp spaces, modes of convergence, Hahn-Jordan decomposition theorem, Riesz representation theorem, Caratheodory and Hahn extension theorems, product measures, outer measures, Caratheodory's theorem, approximation of measurable sets, nonmeasurable and non-Borel sets.
Aims
The aim of the unit is to introduce measure theory and the Lebesgue integral.
Syllabus
Extended Real Number Theory, Measureable Functions, Measures, The Integral, Integrable Functions, Lp spaces, Modes of Convergence, Decompostion of Measures, Generation of Measures, Product Measures, Approximation of Measureable Sets, Non-Borel sets.
Relation to Other Units
This unit is an element of a sequence of Analysis courses at Levels C/4, I/5, H/6 and 7/M. It is a prerequisite for Advanced Topics in Analysis ( which will not run in 2011-2012).
At the end of the course the student should know and understand the definitions and theorems (and their proofs), and should be able to use the ideas of the course in unseen situations.
Transferable Skills:
Assimilation of abstract ideas and reasoning in an abstract context. Setting out a sustained argument in a form comprehensible to others.
A standard lecture course of 30 lectures, 3 revision classes and problem classes.
The final assessment mark for the unit is calculated from a standard 2 ½-hour written closed-book examination in April consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are not permitted.
