Measure Theory

Subject description

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. 

The subject is taught in programs

Objectives and competences

Students acquire basic knowledge of measure theory needed to understand probability  theory, statistics and functional analysis.

Teaching and learning methods

Lectures, exercises, homeworks, consultations

Expected study results

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.

Basic sources and literature

  • 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.

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