The course will exhibit Lebesgue's theory of integration in which integrals can be assigned to a huge range of functions on the real line, thereby greatly extending the notion of integration presented in Prelims. The theory will be developed in such a way that it can be easily extended to a wider framework, but measures other than Lebesgue's will only be lightly touched.
Operations such as passing limits, infinite sums, or derivatives, through integral signs, or reversing the order of double integrals, are often taken for granted in courses in applied mathematics. Actually, they can occasionally fail. Fortunately, there are powerful convergence and other theorems allowing such operations to be justified under conditions which are widely applicable. The course will display these theorems and a wide range of their applications.
This is a course in rigorous applications. Its principal aim is to develop understanding of the statements of the theorems and how to apply them carefully. Knowledge of technical proofs concerning the construction of Lebesgue measure will not be an essential part of the course, and only outlines will be presented in the lectures.
Prof. Charles Batty
Measure spaces. Outer measure, null set, measurable set. The Cantor set. Lebesgue measure on the real line. Counting measure. Probability measures. Construction of a non-measurable set (non-examinable). Simple function, measurable function, integrable function. Reconciliation with the integral introduced in Prelims.
A simple comparison theorem. Integrability of polynomial and exponential functions over suitable intervals. Monotone Convergence Theorem (proof not examinable). Fatou's Lemma (proof not examinable). Dominated Convergence Theorem. Corollaries and applications of the Convergence Theorems (including term-by-term integration of series).
Theorems of Fubini and Tonelli (proofs not examinable). Differentiation under the integral sign. Change of variables.
Brief introduction to \(L^p\) spaces. Hölder and Minkowski inequalities (proof not examinable).