MATH265 : MEASURE
THEORY AND PROBABILITY
AIMS:
To provide a sufficiently deep introduction to the measure
theory and to the Lebesgue theory of integration. To provide a solid
background for the modern probability theory which is essential for
Financial Mathematics.
LEARNING OUTCOMES:
After completing the module students should have a grounding
in the measure theory, measurable functions, Lebesgue integrals and their
properties, they should understand deeply the rigorous foundations of the
modern probability theory and applications to Financial Mathematics.
OUTLINE SYLLABUS:
Set
Theory: set operations (unions, differences, complements),
countable and uncountable sets, Cartesian product, sigma-fields, Monotone
Class Theorem, product sigma-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 Riamann
integrals, Fatou's lemma, Monotone Convergence
Theorem, Dominant Convergence Theorem, convergence in L^p,
Radon-Nikodym Theorem, Riesz
Representation Theorem, conditional expectations, independence, product
measures, Fubini's Theorem.
RECOMMENDED TEXTS:
P.Halmos (1974). Measure Theory.
Springer-Verlag, NY.
M.Taylor (2006). Measure Theory and Integration.
AMS.
M.Capinski and E.Kopp
(2005). Measure, Integral and Probability. Springer-Verlag,
London.
K.Parthasarathy (2005). Probability on Metric
Spaces. AMS
ASSESSMENT WEIGHTINGS: 90% Examination; 10% Continuous Assessment.
MATH RESOURCES:
Basic
formulas
Past exam papers
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