PMAT 42423: Measure Theory

Course Code    : PMAT 42423

Title                   : Measure Theory

Pre-requisites   : PMAT 42363

 Learning Outcomes:

At the end of the course the student should be able to

1. Develop familiarity with measures, Integration theory and convergence as presented in the course

2. Understand the construction of the integral and know its key properties

3. Describe and apply the notion of measurable functions and sets and use Lebesgue monotone and dominated convergence theorems and Fatou’s Lemma

4. Apply theory of integral and understand the definition and basic properties of integral with respect to a measure

5. Analyze and understand how the major theorems presented in the course depend on preliminary results

6. Demonstrate knowledge of the concepts and theorems of abstract Measure Theory and to apply them in Lebesgue integral.

Content:

Sets and classes: 

• Set inclusion, Number sets, Operations on sets, Unions, Intersections, Complement, Differences, Symmetric differences, Monotone sequence of sets, Limits, Inferior and Superior limits, Set of extended real numbers, Countable sets.

Abstract Measure Theory: 

• Algebra, 𝜎-Algebra, Measure, Additivity properties of a set function, Borel sets, Lebesgue measure, Outer measure, Extension of measures ( Measure on a larger class of subsets), Measurable subsets, Measurable functions, Integration with respect to a measure, Properties of integral, Properties that hold almost everywhere, Integrable functions, Additivity theorem, Monotone convergence theorem, Fatou’s lemma, Lebesgue Dominated convergence theorem, Relation between Lebesgue and Riemann integrals, Modes of convergence.

Further results in Measure and Integration: 

• Jordan-Hahn Decomposition theorem, Absolute continuity, Radon Nikodym theorem.

 Method of Teaching and Learning : A combination of lectures, tutorial discussions and presentations.

Assessment : Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading:

1. Cohn, D.L., (2nd edition, 2015) Measure Theory, Springer New York.

2. Barra, G., (2nd edition, 2003) Measure Theory and Integration, Elsevier.