# 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 Rn .
• 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.
• Lp-spaces: inequalities of Jensen, Hölder and Minkovskibounded linear functionals, dual spaces.
• Integration on locally compact spaces: positive linear functionals on Cc(X), Radon measures,  Riesz representation theorem, Lusin’s theorem, density of Cc(X) in Lp-spaces.
• Differentiation of measures on Rn :    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.

### Contact

Main contact:
e-mail: info.stat (at) uni-lj.si

Contact for administrative questions (enrolment, technical questions):
Katarina Erjavec Drešar
University of Ljubljana, Faculty of electrical engineering, Tržaška cesta 25, 1000 Ljubljana.
room num.: AN012C-ŠTU
phone: 01 4768 209
e-mail: katarina.erjavec-dresar (at) fe.uni-lj.si