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Module MA2224: Lebesgue integral
- Credit weighting (ECTS)
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5 credits
- Semester/term taught
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Hilary term 2018-19
- Contact Hours
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11 weeks, 3 lectures per week
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- Lecturer
- Prof David Wilkins
- Learning Outcomes
- On successful completion of this module, students will be able to:
- Discuss countable sets, characteristic functions and bolean algebras;
- State and prove properties of length measure, outer measure and Lebesgue measure for subsets of the real line and establish measurability for a range of functions and sets;
- Define the Lebesgue integral on the real line and apply basic results including convergence theorems.
- Module Content
- The basics of the theory of the Lebesgue integral and Lebesgue measure.Monotone and dominated convergence theorems.
- Countable versus uncountable sets; inverse images;characteristic functions; boolean algebra for subsets.
- Algebras of subsets of the real line; length measure on the interval algebra; finite-additivity; subadditivity and countable-additivity; outer measure; Lebesgue measurable sets; extension to sigma algebra; Borel sigma algebra.
- Lebesgue measurable functions; simple functions; integrals for non-negative functions; limits of measurable functions and the monotone convergence therorem; Lebesgue integrable functions; generalisation of the Riemann integral (for continuous functions on finite closed intervals).
- Fatou's lemma; dominated convergence theorem; integrals depending on a parameter
- Module Prerequisite
- Metric Spaces (MA2223)
- Assessment Detail
- This module will be examined in a 2-hour examination in Trinity term.
Continuous assessment in the form of homework assignments will contribute 10% to the final grade at the annual examinations, with the examination counting for the remaining 90%