Department of Mathematical Sciences, Unit Catalogue 2011/12 

Credits:  6 
Level:  Masters UG & PG (FHEQ level 7) 
Period: 
Semester 1 
Assessment:  EX 100% 
Supplementary Assessment:  Likeforlike reassessment (where allowed by programme regulations) 
Requisites:  Before taking this unit you must take MA20218 and take MA20219 
Description:  Aims & Learning Objectives: Aims: The purpose of this course is to lay the basic technical foundations and establish the main principles which underpin the classical notions of area, volume and the related idea of an integral. Objectives: The objective is to familiarise students with measure as a tool in analysis, functional analysis and probability theory. Students will be able to quote and apply the main inequalities in the subject, and to understand their significance in a wide range of contexts. Students will obtain a full understanding of the Lebesgue Integral. Content: Topics will be chosen from the following: Measurability for sets: algebras, σalgebras, πsystems, dsystems; Dynkin's Lemma; Borel σalgebras. Measure in the abstract: additive and sadditive set functions; monotoneconvergence properties; Uniqueness Lemma; statement of Caratheodory's Theorem and discussion of the lset concept used in its proof; full proof on handout. Lebesgue measure on 3^{n}: existence; inner and outer regularity. Measurable functions. Sums, products, composition, lim sups, etc; The MonotoneClass Theorem. Probability. Sample space, events, random variables. Independence; rigorous statement of the Strong Law for coin tossing. Integration. Integral of a nonnegative functions as sup of the integrals of simple nonnegative functions dominated by it. MonotoneConvergence Theorem; 'Additivity'; Fatou's Lemma; integral of 'signed' function; definition of L^{p} and of L^{p} linearity; DominatedConvergence Theorem  with mention that it is not the `right' result. Product measures: definition; uniqueness; existence; Fubini's Theorem. Absolutely continuous measures: the idea; effect on integrals. Statement of the RadonNikodým Theorem. Inequalities: Jensen, Holder, Minkowski. Completeness of L^{p}. 
Programme availability: 
MA40042 is Optional on the following programmes:Department of Mathematical Sciences
