The main aim is to provide a sufficiently deep introduction to measure theory and to the Lebesgue theory of integration. In particular, this module aims to provide a solid background for the modern probability theory, which is essential for Financial Mathematics.

After completing the module students should be able to:

master the basic results about measures and measurable functions;

to understand deeply the rigorous foundations of probability theory;

to know certain applications of measure theory to probability, random processes, and financial mathematics.

Set Theory: set operations (unions, intersections, differences, complements), countable and uncountable sets, Cartesian product, σ-fields, Monotone Class Theorem, product σ-fields.

Measures: definitions and properties, measurable sets on the straight line, Lebesgue measure, completion, probability measure, probability space.

Measurable functions: definitions of measurable mappings and measurable functions, operations of measurable functions, sequences of measurable functions, almost sure convergence, convergence in measure, random variables, independence.

Integration: definitions and properties, integrable functions, relationship of the Lebesgue and Riemann integrals, Fatou’s lemma, Monotone convergence Theorem, Dominant Convergence Theorem, convergence in L­^p, Radon-Nikodym Theorem, Riesz Representation Theorem,

Measures on metric spaces.

Conditional expectations, independence, product measures, Fubini’s Theorem, filtration, construction of simplest random processes.