**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%.

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