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MA359 Content

Content: The modern notion of measure, developed in the late 19th century, is an extension of the notions of length, area or volume. A measure m is a law which assigns a number  m(A) to certain subsets A of a given space and is a natural generalization of the following notions: 1) length of an interval, 2) area of a plane figure, 3) volume of a solid, 4) amount of mass contained in a region, 5) probability that an event from A occurs, etc.

It originated in the real analysis and is used now in many areas of mathematics like, for instance, geometry, probability theory, dynamical systems, functional analysis, etc.

Given a measure m, one can define the integral of suitable real valued functions with respect to m. Riemann integral is applied to continuous functions or functions with ``few`` points of discontinuity. For measurable functions that can be discontinuous ``almost everywhere'' Riemann integral does not make sense. However it is possible to define more flexible and powerful Lebesgue's integral (integral with respect to Lebesgue's measure) which is one of the key notions of modern analysis.

The Module will cover the following topics: Definition of a measurable space and  \sigma -additive measures, Construction of a measure form outer measure, Construction of Lebesgue's measure, Lebesgue-Stieltjes measures, Examples of non-measurable sets, Measurable Functions, Integral with respect to a measure, Lusin's Theorem, Egoroff's Theorem, Fatou's Lemma, Monotone Convergence Theorem, Dominated Convergence Theorem, Product Measures and Fubini's Theorem. Selection of advanced topics such as Radon-Nikodym theorem, covering theorems, differentiability of monotone functions almost everywhere, descriptive definition of the Lebesgue integral, description of Riemann integrable functions, k-dimensional measures in n-dimensional spaces, divergence theorem, Riesz representation theorem, etc.

Aims: To introduce the concepts of measure and integral with respect to a measure, to show their basic properties, and to provide a basis for further studies in Analysis, Probability, and Dynamical Systems.

Objectives: To gain understanding of the abstract measure theory and definition and main properties of the integral. To construct Lebesgue's measure on the real line and in n-dimensional Euclidean space. To explain the basic advanced directions of the theory.

Books: There is no official textbook for the course. The list below contains some of many books that may be used to complement the lectures.

Stein, E. M. and Shakarchi, R. Real Analysis - measure theory, integration and Hilbert spaces. (Princeton Lectures in Analysis III) Princeton University Press (2005).

Royden, H. L.: Real Analysis, Third Edition, Macmillan Publishing Company (1988).

Rudin, W.: Real and Complex Analysis, Third Edition, McGraw-Hill Book Company (1987).

Halmos, P. R.: Measure Theory, D. Van Nostrand Company Inc., Princeton, N.J. (1950).