# MA2224 : Lebesgue Integral

## Hilary term, 2013

Lecturer: Masha Vlasenko

Lebesgue integral extends Riemann-Darboux itegral in a sense that one can integrate a broader domain of functions. This course is intended to introduce students to the theory of measure and integration. We will construct Lebesgue integral on the real line. The only prerequisites are basic knowlegde of analysis and ability to understand and write proofs.

The course will be accompanied by weekly tutorials. Tutorials take place every Friday at 10 am in SNIAM lecture hall.

Literature
[B] H.S.Bear, A Primer on Lebesgue Integration
[RT] Richard Timoney's lecture notes

3 hour examination in May (joint with metric spaces). The final mark will be 80% of the exam mark + 20% of the continuous assessment mark. The continuous assessment consists of two kinds of work:
• tutorial work: solving a question on the board in class gives 2% (i.e. 1/10 of the maximal 20%). You can't get more then 2% per week even if you solve two questions or more.
• reporter work: writing solutions to a weekly assignment in latex gives 10% (i.e. 1/2 of the maximal 20%). Each week we need a group of up to 5 people ("reporters") to do this job. At the first lecture every week we will make a list of reporters who are responsible to type the solutions and send them to me (preferably in pdf, compiled from latex, though scanned handwriting is also acceptable) by the beginning of the next week. Please sign the sheet with your names. All reporters do one sheet together, and everyone gets 10%. The solutions will be uploaded to this page.

In case you've got more then 20% in the continuous assessment by the end of the course, your mark will be cut down to 20%. One person can be a reporter only once.The questions on the final exam will be close to the ones in the assignments, but also there will be some theoretical questions.

Lecture notes and assignments
1. What is the course about? <pdf>
2. Countable and uncountable sets, inverse images, characteristic functions and Venn diagrams. <pdf>
• read chapter 0 in [RT]
• assignment 1 in <pdf> (please print and bring to class) and <tex> (for reporters)
reporters: Jacob Miller, Kimyra Lynch, Cian Mc Lead, James Larkin, Mark Quinn-Nealon
<solutions> Remarks: in q.4 there is actually no need to introduce the number corresponding to a sequence.
3. Lebesgue outer measure on (0,1).<pdf>
• assignment 2 <pdf> <tex>
reporters: Donal McKeating, Killian O'Donoghue, Michael Toolan, Arianna Fox, Ian Hurley
<solutions>
4. Measurable sets. <pdf> Carathéodory's criterion and set-theoretic operations on measurable sets. <pdf>
• assignment 3 <pdf> <tex>
reporters: Claire MacSharry, Kathleen Keady, Cliona Rogan, Katarina Manojlović, Conor Parle
<solutions>
5. Sigma-algebra of measurable sets. <pdf>
6. Some examples: a non-measurable set, an uncountable set of measure zero. Borel sets. <pdf>
• assignment 4 <pdf> <tex>
reporters: Róisin Basquel, Jennifer Garvey, Ruaidhri O'Dowd, Eoghan O'Neill, Patrick Vaughan
<solutions>
7. The Lebesgue integral for bounded functions. <pdf> Revision. <pdf>
• assignment 5 <pdf> <tex>
reporters: Karl Downey, Eric Hattaway, Graeme Meyler, Conn McCarty, Aran Nolan
<solutions>
8. Properties of the Lebesgue integral. <pdf> Proof of the Bounded Convergence Theorem. <pdf>
• assignment 6 <pdf> <tex>
<solutions>
reporters: Miceal Meenagh, Cian O'Byrne, Bonnie Flattery, Caoimhe Rooney, John Prasifka
9. The integral of unbounded functions, the non-negative case. <pdf>
• assignment 7 <pdf> <tex>
<solutions>
reporters: Garrett Thomas, Erle Holgersen, Padraig Condon, Theo Anderson, Seán Phelan
10. The integral of unbounded functions, the general case. The Lebesgue Dominated Convergence Theorem. <pdf>
• assignment 8 <pdf> <tex>
<solutions> reporters: Lauren Watson, Nastya Vavryk, Ewan Dalby, Sean O'Malley, Keith Glennon
• assignment 9 <pdf> <tex>
<solutions> reporters: Adam Power, Thomas Hogan, Luke Mulcahy, Nancy Rowan Hamilton, John Madden
• assignment 10 <pdf> <tex>
<solutions> reporters: Ciaran Nash, Aisling Mullins, Marianna Kelly
11. An overview of further results: differentiation and integration, absolutely continuous and singular measures, the Lebesgue Decomposition Theorem. <pdf>