| CODE | MAT3217 | ||||||||
| TITLE | Lebesgue Integration | ||||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 6 | ||||||||
| ECTS CREDITS | 4 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | - o - algebras; - Measure spaces; - Dynkin's πλ Theorem - The Measure Extension Theorem - Constructing the Lebesgue measure on ℝn - Measurable functions; - Lebesgue integrable functions; - Convergence theorems; - Theorems of Fubini and Tonelli. Study-Unit Aims: The course will exhibit Lebesgue's theory of integration in which integrals can be assigned to a huge range of functions on ℝn, thereby greatly extending the notion of integration presented in previous calculus courses. Very often, operations such as passing to the pointwise limit through integral signs, or reversing the order of double integrals, are taken for granted in applied mathematics. However, these can occasionally fail. Luckily, there are powerful convergence theorems allowing such operations to be justified under widely applicable conditions. The course will display these theorems and some applications. Learning Outcomes: Knowledge and understanding By the end of the study unit, students will be able to: - Understand the construction and properties of Lebesgue measure, including the notion and properties of null set; - Analyse and explain the connection between integrability as defined in this course, and Riemann integrability; - State, prove and apply the Monotone Convergence Theorem, Fatou's Lemma and the Dominated Convergence Theorem; - State and apply Fubini's Theorem and Tonelli's Theorem. Skills By the end of the study unit, students will be able to: - Understand the fundamental properties of the Lebesgue integral and be able to apply them. Suggested Reading: - Course notes (E. Chetcuti) - Cohn D., Measure Theory, Birkhauser, 1980. - Heinz B., Measure and Integration Theory, de Gruyter Studies in Mathematics 2001 - Fremlin D.H., Measure Theory; Vol. 1: The Irreducible Minimum, Torres Fremlin, 2000. |
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| ADDITIONAL NOTES | Follows from: MAT3215 Leads to: MAT3210 |
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| STUDY-UNIT TYPE | Lecture and Tutorial | ||||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | David Buhagiar |
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The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2024/5. It may be subject to change in subsequent years. |
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