### Module Details

 The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
 Title MEASURE THEORY AND PROBABILITY Code MATH365 Coordinator Dr AB Piunovskiy Mathematical Sciences Piunov@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2021-22 Level 6 FHEQ First Semester 15

### Aims

The main aim is to provide a sufficiently deepintroduction to measure theory and to the Lebesgue theory of integration. Inparticular, this module aims to provide a solid background for the modernprobability theory, which is essential for Financial Mathematics.

### Learning Outcomes

(LO1) After completing the module students should be ableto:

(LO2) master the basic results about measures and measurable functions;

(LO3) master the basic results about Lebesgue integrals and their properties;

(LO4) to understand deeply the rigorous foundations ofprobability theory;

(LO5) to know certain applications of measure theoryto probability, random processes, and financial mathematics.

(S1) Problem solving skills

(S2) Logical reasoning

### Syllabus

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 L­p, Radon-Nikodym Theorem, Riesz Representation Theorem,

Measures on metric spaces.

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

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
final assessment  one hour time on tas    50
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
homework 1  around 60-90 minutes    10
homework 2  around 60-90 minutes    10
Homework 3  around 60-90 minutes    10
Homework 4  around 60-90 minutes    10
homework 5  around 60-90 minutes    10