Introductory description

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.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

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.

Learning outcomes

By the end of the module, students should be able to:

• 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.

Indicative reading list

There is no official textbook for the course. As the main recommended book, I would suggest:
Cohn, D.L, Measure Theory, Second Edition, Birkhauser (2013). *
The list below contains some of many further books that may be used to complement the lectures.
Folland, G.b.: Real Analysis, Second Edition, Wiley (1999). *
Halmos, P. R.: Measure Theory, D. Van Nostrand Company Inc., Princeton, N.J. (1950) (Reprinted by Springer (1974)).
Kubrusly, C.S: Essentials of Measure Theory, Springer (2015). *
Loeb, P.A: Real Analysis, Birkhauser (2016). *
Royden, H. L. and Fitzpatrick, P.M: Real Analysis, Fourth Edition, Macmillan Publishing Company (2010).
Rudin, W.: Real and Complex Analysis, Third Edition, McGraw-Hill Book Company (1987).
Stein, E. M. and Shakarchi, R.: Real Analysis - measure theory, integration and Hilbert spaces. (Princeton Lectures in Analysis III) Princeton University Press (2005).

• = E-book avaliable from Warwick Library.

Subject specific skills

The students will acquire working knowledge of measure and integral with respect to a measure, including sigma-algebras, Dynkin classes, outer measures, the Lebesgue measure on the real line and a finite-dimensional Euclidean space, Lebesgue-Stieltjes measures, measurable and integrable functions, convergence theorems, Lusin's and Egoroff's Theorems, product measures and multivariable integrals, signed measures, Radon-Nikodym theorem, and Lp-spaces. This will form a basis for further studies in Ergodic Theory, Functional Analysis, Probability, etc.

Transferable skills

The students will be able to apply abstract notions in different contexts, in particular they will learn general techniques for studying parameters of large-scale systems arising in physical sciences. The students will also develop the ability to approach complex problems by approximating them by simpler ones.