MATM19 Integration Theory is an optional course for a Master of Science degree in mathematics.
The Riemann integral taught in our beginner courses has the advantage that it is quite easy to define and in many cases easy to compute. At the same time there are many relatively simple functions which are not integrable in this sense, and the integral is quite unstable with respect to pointwise convergent sequences of integrable functions. In this course we present H. Lebesgue's extension of the Riemann integral in the most general context. One major advantage offered by the general approach is that it pertains also to the basics of Probability Theory. The course starts with sigma-algebras and measures on general sets, and the construction of measures based on outer measures. With these notions in hand, it turns to the definition of the integral, examples on the real line like the Lebesgue and Lebesgue-Stieltjes integrals, the Lebesgue integral in higher dimensions, convergence theorems, $L^p$-spaces, iterated integrals and the theorems of Fubini and Tonelli.
The teaching consists of lectures and seminars. Compulsory hand-in exercises may be given.
The course is assessed through a written and an oral examination.
Donald L. Cohn, Measure Theory, 2nd ed, 2013, ISBN: 978-1-4614-6955-1.
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