COURSE OUTLINE

MATH 6280.03   Measure  Theory,     Fall  2001

Time:        Thursday  11:30--2:30
Location:   301 SSB

INSTRUCTOR:     Yun Gao,
Office:                   Ross S624,
Telephone:             736-2100 ext. 33952
e-mail:                   ygao@yorku.ca

OFFICE HOURS:    Tuesdays and Thursdays:  4:00--5:00pm  or  by appointment.

TEXTBOOK:

H.L. Royden,   A Real  Analysis,
third edition,  MacMillan, 1988

TOPICS TO BE COVERED:

Sigma-algebras, measure spaces, measurable functions, outer measure and measurability, extension theorems, integration, convergence theorems, signed measures, Hahn-Jordan decomposition, Radon-Nikodym theorem, product measures, Fubini theorem.

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A measure is a function that assigns numbers to subsets of a given set. For example, the Lebesgue measure of a subset of the plane is the area of the subset for those sets that have a classically defined area, and it also extends the definition of area to a much wider class of subsets. Measures are also important in probability theory, where events correspond to subsets and measures assign probabilities to events. More generally, measure theory is central to analysis, since measures are used to construct integrals and conversely integrals give rise to measures.

This course begins with the classical theory of Lebesgue measure and Lebesgue integration on the real line. We will then examine the general theory of measures on abstract spaces.

Prerequisite:  An undergraduate course in real analysis at the level of the York course  Math 4010.

SYLLABUS:   We will briefly review some basics in Chapters  1 and 2 and then go on to  study the following sections:

Chapter 3:     Lebesgue Measure                          All sections
Chapter 4:     The Lebesgue Integral                     All sections
Chapter 11:   Measure and  Integration                  All sections
Chapter 12:   Measure and Outer Measure             Sections 1--4

GRADING:   There will be  6 assignments which are counted for 60% .  You  are expected to do all of the assigned homework.  There will be a 3-hour-in-class final exam which  is  counted  for  40%.