MA30361: Measure theory and integration
[Page last updated: 22 May 2025]
| Academic Year: | 2025/26 |
| Owning Department/School: | Department of Mathematical Sciences |
| Credits: | 6 [equivalent to 12 CATS credits] |
| Notional Study Hours: | 120 |
| Level: | Honours (FHEQ level 6) |
| Period: |
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| Assessment Summary: | EXCB 100% |
| Assessment Detail: |
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| Supplementary Assessment: |
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| Requisites: | Before taking this module you must take MA22032 OR take MA20218 |
| Learning Outcomes: |
On completing the unit, students should be able to:
芒锟铰emonstrate a good knowledge and understanding of the main results and techniques in measure theory;
芒锟铰emonstrate an understanding of the Lebesgue Integral;
芒锟铰uote 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, Carath锟铰﹐dory'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-Nikod锟铰絤 Theorem. Inequalities of Jensen, H锟铰秎der, Minkowski. Completeness of Lp. For some theorems the proofs will be optional reading material. |
| Course availability: |
MA30361 is Optional on the following courses:Department of Mathematical Sciences
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Notes:
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