Institutions of Advanced Analysis
Learning outcomes of the course unit
Students must demonstrate knowledge and understanding of advanced results of measure theory
and real analysis.
In particular, within the program carried out, students must
1. exhibit solid knowledge and thorough conceptual understanding of the subject;
2. be able to produce rigorous proofs of results related to those examined in the course;
3. be able to evaluate coherence and correctness of results obtained by themselves or by others;
4. learn to formulate and communicate mathematical arguments effectively using the appropriate scientifical lexicon;
5. be able to access autonomously scientific books and articles on the subject.
Previous undergraduate courses in algebra, topology and mathematical analysis.
Course contents summary
Advanced measure theory and real analysis.
CONTENTS (EXTENDED VERSION)
1. Abstract measure theory and integration.
2. Positive Borel measures.
3. Complex measures and Radon-Nikodym theorem.
4. Lp spaces.
5. The fundamental theorem of calculus and AC functions.
6. Differentiaton of measures and integrals in R^N.
7. Hausdorff measures and self similar sets.
W. Rudin, "Real and complex analysis", 3nd Edition, McGraw-Hill Inc., New York 1987;
G. B. Folland, "Real analysis", 2nd Edition, Wiley-Interscience, New York 1999.
Lectures (4 hours per week).
Assessment methods and criteria
Assessment is based on homeworks and a final oral exam.