Back to Tom Salisbury's Home Page
MATH 6280 3.00AF (Fall 2013)
Announcements and documents will be posted here as they
The final exam (and comprehensive)
was held Thursday Dec. 19, 10am-1pm 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
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 Lebesgue-Stieljes measures associated with increasing and right-continuous functions F. The proof has three steps:
The theory this requires is covered in sections 2.2-2.3, 2.5, and 17.3-17.5 of Royden.
- First, construct a pre-measure mu associated with F,
on a concrete algebra of sets.
- Second, extend this pre-measure 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).
- Uniqueness of L-S measures, using monotone classes (not covered in Royden)
- The Cantor measure (section 2.7)
- Non-measurable sets (section 2.6)
- Approximation properties (section 2.4).
- Completions (section 17.1)
- 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,
- Product measure, Fubini and Tonelli (section 20.1)
- Signed measures, and the Jordan-Hahn 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)