Last modified: 22 May 2019 17:07
Measure theory provides a systematic framework to the intuitive concepts of the length of a curve, the area of a surface or the volume of a solid body. It is foundational to modern analysis and other branches of mathematics and physics.
Study Type  Undergraduate  Level  4 

Session  First Sub Session  Credit Points  15 credits (7.5 ECTS credits) 
Campus  None.  Sustained Study  No 
Coordinators 

Syllabus
Extension a measure from a semiring of sets to the corresponding ring of sets;
Algebras and σalgebras of sets;
σadditive and σsemiadditive measures;
Measure spaces;
Lebesgue extension of a measure defined on a ring of sets;
Properties of the Lebesgue measure;
Definition and properties of measurable functions;
Convergence almost everywhere;
Measurable functions and uniform convergence; the Egorov theorem;
Lebesgue integral for simple functions;
The definition and the properties of the Lebesgue integral;
Absolute continuity and σadditivity of the Lebesgue integral; the Chebyshev inequality;
The Lebesgue, the Levi and the Fatou convergence theorems for the Lebesgue integral;
Comparison of the Lebesgue integral and of the Riemann integral.
Course Aims
The aim of the course is to introduce the basic ideas and techniques of measure theory and Lebesgue integration.
Learning Objectives
By the end of the revision period, students should be able (i) to state and illustrate the definitions of the concepts introduced in the course, (ii) to state the theorems of the course, to explain their significance, and to give examples to indicate the role of the hypotheses, (iii) to demonstrate knowledge and understanding of proof techniques used in the course, (iv) to use the methods and results of the course to solve problems at levels similar to those seen in the course.
In particular, students should be able
 to define a &sigma algebra, a measure and a measurable function, to check the definitions in examples and to prove simple results (seen and unseen) about these concepts;
 to integrate (in turn) a simple positive measurable function, a positive measurable function and a general L1 function and derive properties of integrals using this path;
 to evaluate integrals of continuous functions on the real line and to evaluate integrals of sequences with respect to counting measure on the positive integers;
 to state, prove and use the Monotone Convergence Theorem, Fatou’s Lemma and the Dominated Convergence Theorem.
Information on contact teaching time is available from the course guide.
1st Attempt: 1 twohour written examination (80%); incourse assessment (20%). Resit: If required and permitted by Regulations, there will be 1 twohour written examination. The CAS mark will be based on the maximum of examination (100%) and examination (80%) together with incourse assessment (20%).
Informal assessment of weekly homework through discussions in tutorials.
Incourse assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact Course Coordinators for feedback on the final examination.
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