## MA40042: Measure theory & integration

[Page last updated: 03 August 2022]

Owning Department/School: Department of Mathematical Sciences
Credits: 6 [equivalent to 12 CATS credits]
Notional Study Hours: 120
Level: Masters UG & PG (FHEQ level 7)
Period:
Semester 1
Assessment Summary: EX 100%
Assessment Detail:
• Examination (EX 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA20218
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.

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.

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
• 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 Economics
• UHES-AFB04 : BSc(Hons) Economics and Mathematics (Year 3)
• UHES-AAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 4)
• UHES-AKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 4)
• UHES-ACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 4)
Department of Mathematical Sciences
• RSMA-AFM16 : Integrated PhD Statistical Applied Mathematics
• TSMA-AFM17 : MRes Statistical Applied Mathematics
• TSMA-AFM16 : MSc Statistical Applied Mathematics
• USMA-AFB15 : BSc(Hons) Mathematical Sciences (Year 3)
• USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
• USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
• USMA-AFB13 : BSc(Hons) Mathematics (Year 3)
• USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
• USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
• USMA-AFB01 : BSc(Hons) Mathematics and Statistics (Year 3)
• USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
• USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
• USMA-AFB05 : BSc(Hons) Statistics (Year 3)
• USMA-AAB06 : BSc(Hons) Statistics with Study year abroad (Year 4)
• USMA-AKB06 : BSc(Hons) Statistics with Year long work placement (Year 4)
• USMA-AFM14 : MMath(Hons) Mathematics (Year 3)
• USMA-AFM14 : MMath(Hons) Mathematics (Year 4)
• USMA-AAM15 : MMath(Hons) Mathematics with Study year abroad (Year 4)
• USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)
• USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 5)
Department of Physics
• USXX-AFM01 : MSci(Hons) Mathematics and Physics (Year 4)
• USXX-AAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 5)
• USXX-AKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 5)

 Notes: This unit catalogue is applicable for the 2022/23 academic year only. Students continuing their studies into 2023/24 and beyond should not assume that this unit will be available in future years in the format displayed here for 2022/23. Programmes and units are subject to change in accordance with normal University procedures. Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules. Find out more about these and other important University terms and conditions here.