Measure Theory I - Math580

Measure theory is the modern theory of integration, the method of assigning a ''size'' to subsets of an universal set. It is more general, more powerful and more beautiful (though also more technical) than the classical theory of Riemann integration: The course will be a reasonably standard introduction to measure theory, will some emphasis upon geometric aspects. We will cover most of the topics listed below, subject to time and taste:
Riemann Integral, algebra and sigma-algebra, measures, complete measures spaces, elementary properties outer measure, Lebesgue's measure,  measurable functions, Lebesgue integral (simple functions, monotone convergence theorem, Fatou's Lemma, dominate convergence theorem, Fubini's theorem), regularity of measure on metric spaces, Riesz theorem and Lebesgue- Stieljes measure, measure image, L^p-spaces (Holder's inequalities, Minkowski's inequalities), convergence of functions, convergence of measures, change of variables in R^n.  
* Elias M. Stein and Rami Shakarchi, Fourier Analysis. An Introduction, Princeton University Press, Princeton NJ, 2003.
* H. L. Royden, Real Analysis. The Macmillan Company NewYork NY 1968.
* Lawrence C Evans and Ronald F Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, London, UK, 1992.
* Paul Halmos: Measure Theory, D.Van Norstand Company Inc, Princeton NJ, 1964. 

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