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MATH 6280 3.00AF (Fall 2013)
Measure Theory
Announcements and documents will be posted here as they
become available.
Announcements

The final exam (and comprehensive)
was held Thursday Dec. 19, 10am1pm in CB115.
You were responsible for the topics covered in class, with the exception
of conditional expectations and Hausdorff measures.
 Solutions to the 4th assignment are posted on the assignment page [linked to below].
 Textbook: Real Analysis
by Royden and Fitzpatrick; 4th edition. The textbook is optional  you can
follow the course purely using notes from class. I will pull some homework
problems from the text (particularly optional ones), and will announce
sections from the text so that people can read ahead if they wish.
Documents and links
Reserves
Topics
We covered the following topics:
 Definition of a measure, and basic properties (see sections 2.1 and 17.1 of Royden).
 Stated a theorem giving the existence of LebesgueStieljes measures associated with increasing and rightcontinuous functions F. The proof has three steps:
 First, construct a premeasure mu associated with F,
on a concrete algebra of sets.
 Second, extend this premeasure to an outer measure mu* defined
on all sets.
 Third, show countable additivity when we restrict mu* to Borel sets
(actually to what are called the mu*measurable sets).
The theory this requires is covered in sections 2.22.3, 2.5, and 17.317.5 of Royden.
 Uniqueness of LS measures, using monotone classes (not covered in Royden)
 The Cantor measure (section 2.7)
 Nonmeasurable sets (section 2.6)
 Approximation properties (section 2.4).
 Completions (section 17.1)
 Integration
 Measurable functions (sections 3.1, 3.2, 18.1)
 Integration of simple functions
 Integration of positive measurable functions (sections 4.2, 4.3, 18.2)
 Integrable functions (sections 4.4, 18.3)
 Monotone convergence, Fatou's lemma, Dominated convergence
 Convergence in measure and in Lp (sections 5.2, 7.1)
 Minkowski and Holder inequalities, completeness of Lp (sections 7.2,
7.3, 19.1)
 Product measure, Fubini and Tonelli (section 20.1)
 Signed measures, and the JordanHahn decomposition (section 17.2)
 The Radon Nikodym theorem (section 18.4) and applications
in probability (conditional expectations) and analysis.
 Hausdorff measures (you are not responsible for this topic)