# 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.

## 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.

## Stay up to date

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