Mathematics 420/507 (Real Analysis I / Measure Theory and Integration)
Instructor: Prof. I. Laba
Prerequisite: A score of 68% or higher in MATH 321, or equivalent.
- Contact information: Math Bldg 200, 604-822-4457,
- Lectures: MWF 9-10, MATH 202
- Office hours: Mon 1-2, Wed 10-11, Fri 11-12, in MATH 200
- The best way to contact the instructor is by email. Please note that email received on evenings and weekends
will be answered on the next business day. If you cannot attend regular office hours due to schedule conflict,
please make an appointment in advance. Drop-ins and same-day requests for appointments cannot always be accommodated.
Textbook: Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2bd ed.,
John Wiley and Sons, 1999, ISBN 0-471-31716-0
This is a cross-listed 4th year undergraduate and graduate course which develops the
theory of measure and integration. This material is a cornerstone of mathematical analysis
and is an essential part of an advanced mathematical education. It will be used in
functional analysis, harmonic analysis, partial differential equations, probability, mathematical
physics and information theory. The course will be primarily based on the first 3 chapters
of the text:
Your course grade will be based on homework (50%) and the final exam (50%).
Homework problem sets will be assigned roughly biweekly (you should expect 5-6 assignments
in total). Each one will be assigned and posted here at least a week in advance; I will
also send a broadcast email to the class mailing list to alert you to each posted assignment.
To allow for minor absences and short illnesses, the lowest score will be dropped.
Late assignments will not be accepted.
- Measures (Chapter 1): sigma-algebras, outer measures, Borel measures on the real line, Lebesgue measure. We covered essentially everything in this chapter.
- Integration (Chapter 2): measurable functions, integration, convergence theorems, product measures and
- We covered essentially everything in Sections 2.1-2.5, except that we skipped the gamma function in Section 2.3 (p. 58)
- Section 2.6: we discussed the n-dimensional Lebesgue measure and the relation between
Lebesgue and Riemann integration. We skipped the computational part (behaviour of Lebesgue measure under linear transformations, Theorem 2.44, Corollary 2.46 and Theorem 2.47).
- We also skipped Section 2.7, Integration in polar coordinates.
- Differentiation of Measures (Chapter 3): signed measures, Radon-Nikodym theorem, Hardy-Littlewood differentiation,
fundamental theorem of calculus revisited. We will likely be able to cover only the first 2-3 sections.
Updates will be posted here.
The final exam will be 2.5 hour long (standard length at UBC).
The date of the final examination will be announced by the Registar later in the term.
Attendance at the final examination is required, so be careful about
making other committments (such as travel) before this date is confirmed.
The examination will be strictly closed-book:
no formula sheets, calculators, or other aids will be allowed.
[University of British Columbia]