Edinburgh Mathematics Programme

Level 3 Module

323 AaI Analysis and Integration (S2)

Introduction

In this module, two of the fundamental tools of modern mathematical analysis are developed: the basic theory of the Lebesgue integral and metric spaces. Metric spaces were introduced in FoA and here we explore some of the theory more fully. Integration is familiar in practical terms from calculus (and Complex Analysis) but here we look at the theoretical underpinning of the subject. We link metric spaces and integration together in the context of L^p spaces, which are fundamental examples of the function spaces that provide the language for much of modern analysis. We also give applications to Fourier series and related topics. Other applications of the ideas presented occur in probability, topology and partial differential equations.

Aims

1. To construct the Lebesgue integral on the real line up to the Monotone and Dominated Convergence Theorems (with straighforward applications).
2. To develop the theory of metric spaces, including compactness.
3. To construct the L^p spaces as important examples of complete metric spaces and to give applications to Fourier series.

Prerequisites

MS0105 FoA; MS0225 LAG

Syllabus summary

Infinite sets and countability. Construction of the Lebesgue integral and convergence theorems. Metric spaces and compactness. The L^p spaces.

Texts

H. A. Priestley, Introduction to Integration, OUP, QA 308 Pri.

A. J. Weir, Lebesgue Integration and Measure, CUP, QA 312 Wei.

These books cover the Integration part of the module. The former is a much gentler but more wordy introduction to the subject, while the latter is more concise and goes straight to the point. It also contains an appendix on metric spaces.

T. M. Apostol, Mathematical Analysis, 2nd edition, Addison Wesley, 1974, QA 300 Apo. This book has more than you need for the whole module but is a bit dry.

I. J. Maddox, Elements of Functional Analysis, Cambridge, 1970, QA 320 Mad.

W. A. Sutherland, Introduction to Metric and Topological Spaces, Oxford, QA 611.3 Sut.

W. Rudin, Principles of Mathematical Analysis, McGraw Hill, QA 300 Rud.

E.T. Copson, Metric Spaces, Cambridge, QA 613 Cop.

V. Bryant, Metric Spaces: iteration and application, Cambridge, QA 611.28 Bry.

There is no one book on the above list which is recommended above all others.

Syllabus

1. Set theory: infinite unions and intersections. Countability: countability of Q and uncountability of R. (1.5)
2. The Heine-Borel theorem in R. (0.5)
3. Null sets.(1.5)
4. The Lebesgue integral: step functions and their integrals, L^inc. Practical integration -- integrability of continuous functions on closed bounded intervals and the fundamental theorem of calculus. The space of integrable functions L^1 and properties of the integral on it. The convergence theorems and simple applications. (8.5)
5. Metric spaces: definitions and examples (reviewing material from FoA as appropriate.) Open and closed sets, closure, density. Subspaces. Continuity of functions between metric spaces; basic properties (e.g. inverse image of open set is open). Equivalent metrics. (3.5)
6. Compactness: definition of compactness via open coverings. Heine-Borel theorem in R^n. Basic properties (boundedness and attainment of bounds for continuous functions). Sequential compactness (equivalence with compactness not covered in detail). (3)
7. L^p spaces: measurable functions as almost everywhere limits of sequences of step functions. L^p as a vector space with a (semi)-metric. Holder's inequality. Completeness of L^p (as a consequence of the MCT). Applications to Fourier series and analysis. (Possibilities include: the Riemann-Lebesgue lemma; the problem of convergence in L^p: the Riesz-Fischer theorem for L^2; convolution: integral kernels (e.g. Dirichlet, Fejer, Poisson) and Young's inequalities.) (5.5)

Notes and Links

1. The approach taken to Lebesgue integration is via direct construction of the integral and not via measure theory. The advantage of this is that students can be lead to a rigorously defined but practical integral much more quickly and efficiently than with the measure-theoretic approach. A disadvantage of this is that the route through the theory, while direct, is not flexible, and care needs to be taken not to turn up blind alleys via an unhelpful definition, for example. One approach to follow is that of Weir (with one or two simplifications).
2. The section on practical integration should be used to convince the students that the standard techniques of integration -- recognising anti-derivatives, integration by parts and substitution -- are valid in this setting and are consequences of the fundamental theorem of calculus.
3. The MCT and DCT but not Fatou's Lemma should be covered. A basic application to change of order of summation and integration should be included.
4. Linearity of the integral and its construction should be emphasised at all points.
5. In the applications to Fourier series, it is important to consolidate rather than to upset ideas about Fourier series given in earlier courses e.g. FoA, MAM. (In FoA, for example, uniform convergence for Fejer means might well have been treated.) An outing for Fourier transforms is desirable but should not be at the expense of consolidation of material on Fourier series.
6. If time permits, a very brief discussion of integration on R^n and Fubini-Tonelli might be given. Alternatively, links with probability could be pointed out. On the other hand, if time is tight, the applications of L^p spaces could be suitably tailored.
7. The metric space part of the course is intended to be fairly similar to the old Mathematics 3 presentation.

Outcomes

1. Familiarity with countability.
2. Familiarity with null sets in R.
3. Appreciation of the construction of the Lebesgue integral.
4. Ability to use the standard techniques of integral calculus in the context of Lebesgue integration.
5. Ability to use MCT and DCT.
6. Appreciation of topology of metric spaces and continuous functions defined between them.
7. Appreciation of compactness and familiarity with the Heine-Borel theorem and its applications.
8. Familiarity with L^p spaces and their role in Fourier analysis.