Measure and Integration

Overview

- sigma-algebras, measure spaces, measurable functions
- Lebesgue integral, Fatou's lemma, monotone and dominated convergence theorems
- Fubini’s Theorem, change of variables theorem
- Integral inequalities and Lp spaces

Learning Objectives

It is intended that students shall, on successful completion of the module, be able to: understand the concepts of an algebra and a sigma-algebra of sets, additive and sigma-additive functions on algebras of sets, measurability of a function with respect to a sigma-algebra of subsets of the domain, integrability, measure and Lp-convergence of sequences of measurable functions; demonstrate knowledge and confidence in applying the Caratheodory extension theorem, Fatou's lemma and the monotone convergence theorem, the Lebesgue dominated convergence theorem, the Riesz theorem, Fubini’s theorem, change of variable’s theorem and integral inequalities; proofs excepting those of the Caratheodory and Riesz theorems; understand similarities and differences between Riemann and Lebesgue integration of functions on an interval of the real line.

Skills

Analytic argument skills, problem solving, analysis and construction of proofs.

Assessment

None