Lecturer: Keith Ball

Term(s): Term 1

Status for Mathematics students: List A

Commitment: 30 hours

Assessment: 85% Examination, 15% Assignments

Formal registration prerequisites: None

Assumed knowledge:

MA270 Analysis 3 or MA271 Mathematical Analysis 3:

• Uniform Convergence

MA139 Analysis 2 or MA152 Mathematical Analysis 2:

• Riemann Integration
• Uniform Continuity

MA260 Norms, Metrics and Topologies or MA222 Metric Spaces:

• Definition of a topology
• Open and closed sets
• Norms

Useful background: Good knowledge of core analysis courses including material about continuity and convergence.

Synergies: The following modules go well together with Measure Theory:

• MA3G7 Functional Analysis I
• MA3G8 Functional Analysis II
• MA354 Theory of ODEs
• MA350 Introduction to PDEs
• MA3G1 Theory of PDEs

Leads to: The following modules have this module listed as assumed knowledge or useful background:

• MA3G8 Functional Analysis II
• MA3H2 Markov Processes and Percolation Theory
• MA3D4 Fractal Geometry
• MA3K1 Mathematics of Machine Learning
• MA433 Fourier Analysis
• MA4L2 Statistical Mechanics
• MA4A2 Advanced Partial Differential Equations
• MA427 Ergodic Theory
• MA4F7 Brownian Motion
• MA4J0 Advanced Real Analysis
• MA482 Stochastic Analysis
• MA4L3 Large Deviation Theory
• MA4L9 Variational Analysis and Evolution Equations
• MA4M9 Mathematics of Neuronal Networks

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 from 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, Fatou's Lemma, Monotone Convergence Theorem, Dominated Convergence Theorem, Product Measures and Fubini's Theorem. Selection of advanced topics such as Radon-Nikodym theorem, L_p spaces, 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. 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 available from Warwick Library.

Additional Resources