Course Description
Concept of measure, Lebesgue outer measure and its properties, -algebra, Borel algebra, measurable sets, non-measurable sets, Cantor set, concept and properties of measurable functions, operations on measurable functions, Littlewood’s three principles, construction of Lebesgue integration, simple functions, the space L1 of integrable functions, Monotone convergence theorem and Lebesgue dominated convergence theorem, Fubini theorem,L^p spaces, product measure.
Course Objectives & Outcomes
Objectives:
- Identify measure, measurable set and non-measurable set.
- Identify measurable functions and some of their properties.
- Investigate Lebesgue integrability of any measurable function using convergence theorems
- Distinguish the difference between the Riemann integral and Lebesgue integral.
- Identify some important inequalities and properties of Lebesgue spacesL^p.
- Mention product measure and Fubini-Tonilli theorem
Outcomes:
- Upon successful completion of this course, the student will be able to:
- Define concepts such as measure, measurable set, measurable functions and their algebraic properties.
- Discuss using dominated converging and monotone converging theorems to check the ability for integration.
- Different between the Riemann integral, the Lebesgue integral.
- Outline some properties of Lebesgue spacesL^p.
- Mention product measure and Fubini-Tonilli theorem.
References
1. W. Royden, and P. M. Fitzpatrick " Real Analysis", John Wiley &Sons 4th edition, 2010. ISBN: 0135113555, 9780135113554.
2. E. M. Stein and Rami Shakarchi, “Real Analysis: Measure Theory, Integration, and Hilbert Spaces”, Princeton University press, 2005. ISBN: 9780691113869.ISBN10:0691113866
3. Carlos S. Kubrusly, Measure Theory: A First Course, Academic Press; 1 edition, 2006. ISBN-13: 978-0123708991, ISBN-10: 0123708990.
4. Robert G. Bartle, “The Elements of Integration and Lebesgue Measure”, Wiley-Interscience,1995, ISBN: 0-471-04222-6 ISBN9780471042228
5. De Barra, G.,1981,Measure Theory And Integration. Chichester [England]: E. Horwood,.ISBN: 0 8522618615678910.
M.A.Al-Gwaiz and S.A. Elsanousi, Elements of real analysis , Chapman &Hall /CRC, 2007. ISBN 978-1-58488661-7.
Course ID: MATH 507
Credit hours | Theory | Practical | Laboratory | Lecture | Studio | Contact hours | Pre-requisite | 3 | 3 | 3 | Real Analysis II |
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