Course Overview: 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.
Learning Outcomes: By the end of the course, students will be able to
\(\bullet\) use the definition of a measure space and prove results using this definition, including about key examples of measures
\(\bullet\) use the definitions of measurable functions and integrable functions and prove results using these definitions
\(\bullet\) analyse and explain the connection between integrability as defined in this course, and Riemann integrability from Prelims Analysis III
\(\bullet\) state, prove and apply the Monotone Convergence Theorem, Fatou's Lemma and the Dominated Convergence Theorem
\(\bullet\) state and apply Fubini's Theorem and Tonelli's Theorem
\(\bullet\) use the definition of \(L^p\)-spaces and prove results about them
Course Synopsis: 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. Fatou's Lemma. 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.
