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Measure theory

Financial Mathematics, Second cycle
1 ali 2 year
first or second
slovenian, english
Hours per week – 1. or 2. semester:

There are no prerequisites.

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.


C. D. Aliprantis, O. Burkinshaw: Principles of Real Analysis, 3rd edition, Academic Press, San Diego, 1998.
R. Drnovšek: Rešene naloge iz teorije mere, DMFA-založništvo, Ljubljana, 2001.
G. B. Folland: Real Analysis : Modern Techniques and Their Applications, 2nd edition, John Wiley & Sons, New York, 1999.
M. Hladnik: Naloge in primeri iz funkcionalne analize in teorije mere, DMFA-založništvo, Ljubljana, 1985.
S. Kantorovitz: Introduction to Modern Analysis, Oxford Univ. Press, 2003.
B. Magajna: Osnove teorije mere, DMFA-založništvo, Ljubljana, 2011.
W. Rudin: Real and Complex Analysis, 3rd edition, McGraw-Hill, New York, 1987.

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 of the graduate study of mathematics since it is needed in other areas, for example, in probablity calculus, statistics and functional analysis. It is useful also in other sciences, for example in economy.
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.

Learning and teaching methods

Lectures, exercises, homeworks, consultations


2 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Roman Drnovšek:
DRNOVŠEK, Roman. Spectral inequalities for compact integral operators on Banach function spaces. Mathematical proceedings of the Cambridge Philosophical Society, ISSN 0305-0041, 1992, let. 112, str. 589-598. [COBISS-SI-ID 8169561]
DRNOVŠEK, Roman. On invariant subspaces of Volterra-type operators. Integral equations and operator theory, ISSN 0378-620X, 1997, let. 27, št. 1, str. 1-9. [COBISS-SI-ID 7038553]
DRNOVŠEK, Roman. A generalization of Levinger's theorem to positive kernel operators. Glasgow mathematical journal, ISSN 0017-0895, 2003, vol. 45, part 3, str. 545-555. [COBISS-SI-ID 12825945]
Marko Kandić:
KANDIĆ, Marko. On algebras of polynomially compact operators. Linear and Multilinear Algebra. 2016, vol. 64, no. 6, str. 1185-1196. [COBISS-SI-ID 17493337]
DRNOVŠEK, Roman, KANDIĆ, Marko. Positive operators as commutators of positive operators. Studia Mathematica. 2019, tom 245, str. 185-200. [COBISS-SI-ID 18407769]
KANDIĆ, Marko, MARABEH, Mohammad A. A., TROITSKY, Vladimir G. Unbounded norm topology in Banach lattices. Journal of mathematical analysis and applications. [Print ed.]. July 2017, vol. 451, iss. 1, str. 259-279. [COBISS-SI-ID 18160729]