School of Mathematics and Statistics (SoMaS)

MAS451 Measure and Probability

The module will give students an additional opportunity to develop skills in modern analysis as well as providing a rigorous foundation for probability theory. In particular it would form a companion course to MAS436 (Functional Analysis) and MAS452 (Stochastic Processes and Finance), the latter of which is fundamentally dependent on measure theoretic ideas

Prerequisites: MAS221 (Analysis)
Not with: MAS350 (Measure and Probability)
No other modules have this module as a prerequisite.

Outline syllabus

• The scope of measure theory,
• σ-algebras,
• Properties of measures,
• Measurable functions,
• The Lebesgue integral,
• Interchange of limit and integral,
• Probability from a measure theoretic viewpoint,
• Characteristic functions,
• The central limit theorem.

Aims

• give a more rigorous introduction to the theory of measure.
• develop the ideas of Lebesgue integration and its properties.
• recall the concepts of probability theory and consider them from a measure theoretic point of view.
• prove the Strong Law of Large Numbers and the Central Limit Theorem using these methods.

Learning outcomes

• understand why a more sophisticated theory of integration and measure is needed;
• show that certain functions are measurable;
• construct the Lebesgue integral;
• understand properties of the Lebesgue integral;
• develop probabilistic concepts (random variables, expectation and limits) within the framework of measure theory.
• show a wider understanding of some aspects of measures that are dealt with less fully on the level 3 course, such as product measures and Fubini’s theorem.

Teaching methods

There will be two formal lectures per week, which will involve the formulation of new theory and worked examples. Students will work through set exercises, and also submit homework for marking, although this will not be part of the formal assessment.

Assessment

One 2 hours 30 minutes exam.

Reading list