(in Polish) Measure Theory
1000-OG-EN-TM
Lectures
Constructions and extensions of measures
The Lebesgue integral
The Radon–Nikodym theorem
Images of measures under mappings; change of variables
Disintegration theorem; Young measures
The Fourier transform, Bochner theorem, Bernstein functions
Hausdorff measures, Hausdorff dimension, Choquet capacities,
Differentiation and the Integration, Egoroff theorem, Vitali Covering Theorem, Hardy and Littlewood inequality
The area and coarea formulas
Density points and Lebesgue points
Uniform integrability, Komlos theorem, Dunford-Pettis theorem, Lebesgue–Vitali theorem
Convergence of measures; Kantorovich–Rubinshtein metric, Wasserstein metric, Levy–Prohorov metric, biting lemma
Lorentz spaces, Marcinkiewicz spaces, BMO spaces
The Skorochod space, Wiener measure, Gaussian measure, Hermite polynomials
Classes
Applications of abstract theorems in practice.
Total student workload
Contact hours with teacher:
- participation in lectures and classes - 60 hrs
Self-study hours:
- preparation for lectures - 20 hrs
- preparation for classes – 20 hrs
- preparation for test - 25 hrs
- preparation for examination- 35 hrs
Altogether: 160 hrs ( 6 ECTS)
Learning outcomes - knowledge
Student
W1: has an advanced knowledge of the basic branches of mathematics, K_W01,
W2: has a good understanding of the role and importance of the
methods and notions of measure theory, K_W02,K_W03
W3: knows the connections of the measure theory with other branches of theoretical and applied mathematics, K_W04
Learning outcomes - skills
Student
K_U01 has the skill of constructing mathematical reasonings: proving theorems and finding counterexamples
K_U07 in the measure theory is able to construct complex proofs that require the use of tools from other branches of mathematics.
K_U08 is able to accurately formulate questions/problems to deepen his/her own understanding of a given subject or to find missing elements of reasonings.
Learning outcomes - social competencies
Student:
K1: knows the limitations of his/her own knowledge and skills related to mathematical methods, K_K03
K2: can formulate his/her own opinions related to some topics of contemporary mathematics, has the ability to work systematically on any long-term projects, K_K04
Teaching methods
Expository teaching methods:
- informative lecture
- classes
Prerequisites
Basic knowledge of notions and methods of mathematical analysis, functional analysis, differential equations, algebra, geometry and topology and probability.
Course coordinators
Assessment criteria
Additional information
Additional information (registration calendar, class conductors,
localization and schedules of classes), might be available in the USOSweb system: