Academic Year:
 2015/6 
Owning Department/School:
 Department of Mathematical Sciences 
Credits:
 6 
Level:
 Masters UG & PG (FHEQ level 7) 
Period: 
Semester 1

Assessment Summary:
 EX 100% 
Assessment Detail:  
Supplementary Assessment: 
MA40042 Mandatory extra work (where allowed by programme regulations)

Requisites: 
Before taking this module you must take MA20218 AND take MA20219

Description:
 Aims: 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. To familiarise students with measure as a tool in analysis, functional analysis and probability theory.
Learning Outcomes: On completing the course, students should be able to:
* demonstrate a good knowledge and understanding of the main results and techniques in measure theory;
* demonstrate an understanding of the Lebesgue Integral;
* quote and apply the main inequalities of measure theory in a wide range of contexts.
Skills: Numeracy T/F A
Problem Solving T/F A
Written and Spoken Communication F (in tutorials).
Content: Systems of measurable sets: σalgebras, πsystems, dsystems, Dynkin's Lemma, Borel σalgebras. Measure in the abstract: convergence properties, Uniqueness Lemma, Carathéodory's Theorem (statement). Lebesgue outer measure and measure on R^{n}. Measurable functions. MonotoneClass Theorem. Probability. Random variables. Independence. Integration of nonnegative and signed functions. MonotoneConvergence Theorem. Fatou's Lemma. DominatedConvergence Theorem. Expectation. Product measures. Tonelli's and Fubini's Theorem. RadonNikodým Theorem (statement). Inequalities of Jensen, Hölder, Minkowski. Completeness of L^{p}.

Programme availability: 
MA40042 is Optional on the following programmes:
Department of Computer Science
 USCMAFM14 : MComp(Hons) Computer Science and Mathematics (Year 4)
 USCMAAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 5)
 USCMAKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 5)
Department of Mathematical Sciences
 USMAAFB15 : BSc(Hons) Mathematical Sciences (Year 3)
 USMAAAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
 USMAAKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
 USMAAFB13 : BSc(Hons) Mathematics (Year 3)
 USMAAAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
 USMAAKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
 USMAAFM14 : MMath(Hons) Mathematics (Year 3)
 USMAAFM14 : MMath(Hons) Mathematics (Year 4)
 USMAAAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
 USMAAKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
 USMAAKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)
 USMAAFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
 USMAAAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
 USMAAKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
 TSMAAFM09 : MSc Mathematical Sciences
 TSMAAPM09 : MSc Mathematical Sciences
 TSMAAFM08 : MSc Modern Applications of Mathematics
 TSMAAWM14 : MSc Modern Applications of Mathematics
 USMAAFB05 : BSc(Hons) Statistics (Year 3)
 USMAAAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
 USMAAKB06 : BSc(Hons) Statistics with Year long work placement (Year 4)
Department of Physics
 USXXAFM01 : MSci(Hons) Mathematics and Physics (Year 4)
 USXXAAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 5)
 USXXAKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 5)
