Measure Theory 1000-135TM
- https://moodle.mimuw.edu.pl/course/view.php?id=2241 (term 2024Z)
Program of the lecture:
1. Outer measure, measurable sets, sigma algebra of measurable sets.
2. Definition and properties of the outlet measures: regular, Borel, Borel regular, Radon, restriction.
3. Measurable functions and their properties, theorem of Luzin and Jegorow.
4. Convergence almost everywhere and in measure, their relations
5. Integral and limits: Fatou's lemma, monotone convergence theorem, dominated convergence lemma, Vittali lemma.
6. Integrability and summability, definition of the integral for nonnegative function, and in general, Bochner integral.
7. Signed measures, complex measures, vector measures, basics properties, Hahn decomposition, Jordan decomposition, total variation of measure, bounded Radon measure as a Banach space
8. Covering theorems: Vitali and Besicovitch's
9. Radon-Nikodym derivative, and their characterizaton, canonical decomposition into absolutely continuous and singular part, approximative limit and approximative continuity.
10. Product measure and Fubini theorem
11. Slicing lemma and its applications
12. Hausdorff measure and isodiametric inequality
13. Maximal function, theorem of Hardy-Littlewood.
14. Riesz representation theorem
15. Weak convergence and compactness for bounded Radon measures
Main fields of studies for MISMaP
Type of course
Mode
Prerequisites (description)
Course coordinators
Term 2024Z: | Term 2023Z: |
Learning outcomes
A student:
1. knows the notions of the outer measure, a measurable set, sigma-alagebra of measureable sets and properties and knows how to use these notions.
2. knows the notions of regular, Borel outer measures, Radon measures and know how to apply them.
3. knows ho to approximate sets by open or closed sets for a given Radon measure; knows the Caratheodory characterization of Borel outer measures.
4. knows the notion of measurable functions, knows how to apply Egorova na Lusin Theorems.
5. knows the notions of the integral and knows how to apply it.
6. Knows theorems permitting limit ppassages under the integral sign.
7. knows the notion of real and vector measures, and the total variation.
8. knows the covering theorems and knows how to apply them.
9. knows the Radon-Nikodym derivative
10. knows the product measures and Fubini theorem.
11. knows the Hausdorff measures and its properties.
Assessment criteria
exam
Bibliography
1. L. Evans R. Gariepy: Measure Theory and Fine Properties of Function, CRC Press, 1992
2. H. Federer Geometric: Measure Theory, Springer-Verlag, New York, 1969.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
- Bachelor's degree, first cycle programme, Mathematics
- Master's degree, second cycle programme, Mathematics
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: