On successful completion of this module, students will be able to:
Discuss countable sets, characteristic functions and bolean algebras;
State and prove properties of length measure, outer measure and Lebesgue measure for subsets of the real line and establish measurability for a range of functions and sets;
Define the Lebesgue integral on the real line and apply basic results including convergence theorems.
The basics of the theory of the Lebesgue integral and Lebesgue measure.Monotone and dominated convergence theorems.
Countable versus uncountable sets; inverse images;characteristic functions; boolean algebra for subsets.
Algebras of subsets of the real line; length measure on the interval algebra; finite-additivity; subadditivity and countable-additivity; outer measure; Lebesgue measurable sets; extension to sigma algebra; Borel sigma algebra.
Lebesgue measurable functions; simple functions; integrals for non-negative functions; limits of measurable functions and the monotone convergence therorem; Lebesgue integrable functions; generalisation of the Riemann integral (for continuous functions on finite closed intervals).
Fatou's lemma; dominated convergence theorem; integrals depending on a parameter
Metric Spaces (MA2223)
This module will be examined jointly in a 2-hour examination in Trinity term. Continuous assessment in the form of weekly tutorial work will contribute 20% to the final grade at the annual examinations, with the examination counting for theremaining 80%. For supplementals, if required, the supplemental exam will count 100%.