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MA40042: Measure theory & integration

Follow this link for further information on academic years Academic Year: 2016/7
Further information on owning departmentsOwning Department/School: Department of Mathematical Sciences
Further information on credits Credits: 6      [equivalent to 12 CATS credits]
Further information on notional study hours Notional Study Hours: 120
Further information on unit levels Level: Masters UG & PG (FHEQ level 7)
Further information on teaching periods Period:
Semester 1
Further information on unit assessment Assessment Summary: EX 100%
Further information on unit assessment Assessment Detail:
  • Examination (EX 100%)
Further information on supplementary assessment Supplementary Assessment:
MA40042 Mandatory extra work (where allowed by programme regulations)
Further information on requisites Requisites: Before taking this module you must take MA20218 AND take MA20219
Further information on descriptions 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: σ-algebras, π-systems, d-systems, Dynkin's Lemma, Borel σ-algebras. Measure in the abstract: convergence properties, Uniqueness Lemma, Carathéodory's Theorem (statement). Lebesgue outer measure and measure on Rn. Measurable functions. Monotone-Class Theorem. Probability. Random variables. Independence. Integration of non-negative and signed functions. Monotone-Convergence Theorem. Fatou's Lemma. Dominated-Convergence Theorem. Expectation. Product measures. Tonelli's and Fubini's Theorem. Radon-Nikodým Theorem (statement). Inequalities of Jensen, Hölder, Minkowski. Completeness of Lp.
Further information on programme availabilityProgramme availability:

MA40042 is Optional on the following programmes:

Department of Computer Science
  • USCM-AFM14 : MComp(Hons) Computer Science and Mathematics (Year 4)
  • USCM-AAM14 : MComp(Hons) Computer Science and Mathematics with Study year abroad (Year 5)
  • USCM-AKM14 : MComp(Hons) Computer Science and Mathematics with Year long work placement (Year 5)
Department of Mathematical Sciences Department of Physics

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