MATH4MI Measure and Integration
In school and in introductory courses in mathematics, integral usually means Riemann integral of a real-valued function on the real line. This fundamental concept reveals its beauty in the Fundamental Theorem of Calculus which relates integration and derivation. However, the Riemann integral has some shortfalls that make it inadequate for many purposes in modern analysis. One of them is a gap in the fundamental theorem of calculus: the class of Riemann integrable functions does not coincide with the class of all functions that have an antiderivative. Another drawback is that the interchange of point-wise limits of function sequences and their integrals is only possible under rather restrictive conditions.
In this paper we introduce a modern theory of integration via measure theory that overcomes these shortfalls. It goes back to the French mathematician Henri Léon Lebesgue (1875–1941) and is the gate to many exciting branches of mathematics, like, for instance, modern probability theory, functional analysis, or the theory of partial differential equations. Topics include sigma-algebras, uniqueness and existence of measures, measurable mappings, the construction of the Lebesgue integral, convergence theorems and basic function spaces.
2021, Semester 1.
None. MATH201 and MATH301 are strongly recommended.
Timothy Candy (email@example.com)
To be announced
- R. L. Schilling, Measures, Integrals and Martingales, Cambridge University Press, 2005.
- D. L. Cohn, Measure Theory, Second Edition, Birkhäuser, 2013.
- N. V. Krylov, Introduction to the Theory of Difussion Processes, American Mathematical Society, 1995.
- W. Rudin, Real and Complex Analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill, New York, 1974.
- M. E. Munroe, Measure and Integration, Addison-Wesley Publishing Company, Inc., 1953.
- G. B. Folland, Real Analysis, 2nd. Ed., Wiley, 1999.
Your final mark F in the paper will be calculated according to this formula:
F = max(E, 0.4E + 0.6A)
- E is the Exam mark
- A is the Assignments mark
and all quantities are expressed as percentages.
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