# 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 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 -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).