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MA32061: Measure theory and integration

[Page last updated: 22 May 2025]

Academic Year: 2025/26
Owning Department/School: Department of Mathematical Sciences
Credits: 5 [equivalent to 10 CATS credits]
Notional Study Hours: 100
Level: Honours (FHEQ level 6)
Period:
Semester 2
Assessment Summary: EXCB 100%
Assessment Detail:
  • Closed-book written examination (EXCB 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA22032 OR take MA20218
Learning Outcomes: On completing the unit, 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.



Synopsis: This unit introduces you to the mathematical concept of a measure, which is an abstract way of defining length/area/volume. This concept will allow you to answer questions such as "what is the length (measure) of the set of rational numbers?", and also allow you to define rigorously the probability of an event occurring. This unit will then apply these concepts to develop the Lebesgue integral - a more powerful form of integration than the Riemann integral that you learnt about in Analysis 2A.

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. Lebesgue measure on Rn. Measurable functions. Monotone-Class Theorem. Lebesgue integration. Probability, Random variables, Expectation. Monotone-Convergence Theorem. Fatou's Lemma. Dominated-Convergence Theorem. Product measures. Tonelli's and Fubini's Theorem. Radon-Nikodým Theorem. Inequalities of Jensen, Hölder, Minkowski. Completeness of Lp. For some theorems the proofs will be optional reading material.

Course availability:

MA32061 is Optional on the following courses:

Department of Mathematical Sciences
  • 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-AFB30 : 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-AFB32 : 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-AFM30 : MMath(Hons) Mathematics (Year 3)
  • USMA-AKM15 : MMath(Hons) Mathematics with Year long work placement (Year 4)

Notes:

  • This unit catalogue is applicable for the 2025/26 academic year only. Students continuing their studies into 2026/27 and beyond should not assume that this unit will be available in future years in the format displayed here for 2025/26.
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