Course: Measure Theory
Course type: compulsory
Lecturer: Bojan Magajna, Ph.D., Full Professor
Study programme and level | Study field | Academic year | Semester |
---|---|---|---|
Applied statistics, second level | Mathematical statistics | 1st | 1st |
For the timeline see Curriculum.
Prerequisites:
- Enrolment into the first year of the programme.
Content (Syllabus outline):
- Measures: σ-algebras, positive measures, outer measures, Caratheodory’s theorem, extension of measures from algebras to σ-algebras, Borel measures on R, Lebesgue measure on R.
- Measurable functions: approximation by step functions, modes of convergence of sequences of functions, Egoroff’s theorem.
- Integration: integration of nonnegative functions, Lebesgue monotone convergence theorem, Fatou’s lemma, integration of complex functions, Lebesgue dominated convergence theorem, comparison with Riemann’s integral.
- Product measures: construction of product measures, monotone classes, Tonelli’s and Fubini’s theorem, the Lebesgue integral on R^{n }.
- Complex measures: signed measures, the Hahn and the Jordan decomposition, complex measures, variation of a measure,
- absolute continuity and mutual singularity, the Lebesgue-Radon-Nikodym theorem.
- L^{p}-spaces: inequalities of Jensen, Hölder and Minkovski, bounded linear functionals, dual spaces.
- Integration on locally compact spaces: positive linear functionals on C_{c}(X), Radon measures, Riesz representation theorem, Lusin’s theorem, density of C_{c}(X) in L^{p}-spaces.
- Differentiation of measures on R^{n} : differentiation of measures, absolutely continuous and functions of bounded variation
Objectives and competences:
Students acquire basic knowledge of measure theory needed to understand probability theory, statistics and functional analysis.
Intended learning outcomes:
- Knowledge and understanding: understanding basic concepts of measure and integration theory.
- Application: measure theory is a part of the basic curriculum since it is crucial for understanding the theoretical basis of probablity and statistics.
- Reflection: understanding of the theory on the basis of examples of application.
- Transferable skills: Ability to use abstract methods to solve problems. Ability to use a wide range of references and critical thinking.