|
 Academic Year: |
2014/5 |
| 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: |
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| Supplementary Assessment: |
MA40042 Mandatory extra work (where allowed by programme regulations) |
| Requisites: |
Before taking this unit 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: 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. |
| Programme availability: |
MA40042 is Optional on the following programmes:Department of Computer Science
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