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Trinity College Dublin

TCD Mathematics

School of Mathematics

Mathematics MA2224, Lebesgue integral

Notes

Some (not all) parts of the notes will be in the form of a handout or will be available here. All will be in PDF format and require a programme such as Adobe Acrobat Reader to read them.

Chapter 0, Introduction
Introductory remarks, a very brief overview of what we are aiming to do.
Chapter 1, Some requisites
Some things we need later: countable versus uncountable sets; inverse images; characteristic functions; boolean algebra for subsets.
Chapter 2, Length measure on R
Algebras of subsets of the 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.
Chapter 3, Lebesgue integral and the monotone convergence theorem
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).
Chapter 4, Lebesgue dominated convergence theorem and applications
Fatou's lemma; dominated convergence theorem; integrals depending on a parameter; almost everywhere.