Lectures:
Monday 8:00-9:30, HS 3.
Friday 13:15-14:45, HS 13.

Office hours:
Friday 15:00-16:00, Office 09.135.

Description:
The theory of measure and integration belongs to the foundations of modern analysis, and provides the formal framework for probability theory. This course introduces its central concepts and results (discussing in particular: existence, uniqueness, basic properties, and examples of measures; the abstract Lebesgue integral, convergence theorems, and spaces of integrable functions; product measures; measures with densities; applications to real analysis), and offers brief appetizers for some more advanced topics.

Assessment: Written exam (Monday, Feb 05, in HS3).

Bibliography:
[Ax] Sheldon Axler, Measure, Integration & Real Analysis, Graduate Texts in Mathematics (available online).
[Ba] Richard F. Bass, Real Analysis for Graduate Students, Version 4.3 (available online).
[SS] Elias M. Stein and Rami Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton Lectures in Analysis.

Announcements

• There is no office hour on Friday 14 Oct. The lecture will be given by Prof. Bruin.
• Lectures on Monday are moved to HS 3.
• The final exam has been scheduled on Monday 05th February in HS 3.

Lesson Register

• Mon 02/10: recap on Riemann integrals ([Ax, Chapter 1]); preliminaries ([Ba, Chapter 1]).
• Fri 06/10: construction of Vitali set ([Ba, Sec. 4.4]); sigma-algebras and the Monotone Class Theorem ([Ba, Chapter 2]).
• Mon 09/10: definition and properties of measures ([Ba, Chapter 3]).
• Fri 13/10: outer measures and Caratheodory's Theorem ([Ba, Chapter 4.1]).
• Mon 16/10: properties of Lebesgue-Stieltjes measures ([Ba, Chapter 4.2]).
• Fri 20/10: examples of Lebesgue-Stieltjes measures ([Ba, Chapter 4.3]); Caratheodory's Extension Theorem ([Ba, Chapter 4.5]).
• Mon 23/10: Bernoulli measures (notes); Lebesgue Density Theorem (notes).
• Fri 27/10: measurable functions ([Ba, Chapter 5.1]).
• Mon 30/10: example of a Lebesgue measurable set that is not Borel measurable ([Ba, Chapter 5.1]); approximation by simple functions, Egorov's Theorem ([Ax, Chapter 2E]).
• Fri 03/11: Lusin's Theorem ([Ax, Chapter 2E]); various definitions of the integral of a nonnegative measurable function, integrals of simple functions ([Ax, Chapter 3A]).
• Mon 06/11: the Monotone Convergence Theorem; integral of a measurable function: definition and basic properties ([Ax, Chapter 2E]).
• Fri 10/11: Fatou's Lemma ([Ba, Chapter 7.3]); the Lebesgue Dominated Convergence Theorem ([Ba, Chapter 7.4]).
• Mon 13/11: criteria for a function to be zero almost everywhere ([Ba, Chapter 8.1]); comparison between Riemann and Lebesgue integrals ([Ba, Chapter 9]).
• Fri 17/11: types of convergence ([Ba, Chapter 10]).
• Mon 20/11: product sigma-algebras and product measures ([Ba, Chapter 11.1]).
• Fri 24/11: the Borel sigma algebra B(R^2) in R^2 coincides with the product B(R)xB(R) of the Borel sigma algebras in R ([Ax, Chapter 5C]); the Fubini-Tonelli Theorem ([Ax, Chapter 5B] or [Ba, Chapter 11.2-3]).
• Mon 27/11: signed measures, the total variation of a signed measure ([Ax, Chapter 9A pp. 255-262]); positive and negative sets ([Ba, Chapter 12.1]).
• Fri 01/12: Hahn's Decomposition Theorem ([Ba, Chapter 12.2]); Jordan's Decomposition Theorem ([Ba, Chapter 12.3]).
• Mon 04/12: Radon-Nikodym's and Lebesgue's Decomposition Theorems ([Ba, Chapter 13]); Vitali's Covering Lemma ([Ax, Chapter 4A]).
• Fri 08/12: holiday (no lecture).
• Mon 11/12: the Hardy-Littlewood Maximal Inequality and the Lebesgue Differentiation Theorem ([Ax, Chapter 4A-B], or [Ba, Chapter 14.1-2] for the general case of R^n).
• Fri 15/12: definition of L^p spaces; Holder and Minkowski's Inequalities ([Ba, Chapter 15.1]).
• Mon 08/01: L^p spaces are Banach; approximation by continuous functions ([Ba, Chapter 15.2]).
• Fri 12/01: convolutions and mollifications (notes).
• Mon 15/01: dual spaces ([Ba, Chapter 15.4]).
• Fri 19/01: introduction to Sobolev spaces (notes).
• Mon 22/12: mock exam (script).
• Fri 26/12: correction of the mock exam.
• Mon 29/12: no lecture.