MAT 324, Real Analysis (Measure Theory)

Fall 2011

Christopher Bishop

Professor, Mathematics
SUNY Stony Brook

Office: 4-112 Mathematics Building
Phone: (631)-632-8274
Dept. Phone: (631)-632-8290
FAX: (631)-632-7631

Time and place: MWF 11:45-12:40, Phyics P-127

We will follow the text `Measure, Integral and Probability' by Marek Capinski and Ekkehard Kopp (Springer-Verlag, Springer Undergraduate Mathematics Series, ISBN 1-85233-781-8). I hope to the entire book, at a rate of about 1 chapter every two weeks.

This is definitely a course with proofs. Homework problems will be asssigned for each section and there will be an in-class midterm and a final.

Please hand homework in on or before due date. I will try to discuss the problems on the following meeting. Incorrect problems may be rewritten and handed back in for partial credit.

Hugh Woodin, The Continuum Hypothesis, Part I

This gives an introduction to set theory with a discussion of the the role of the axiom of choice and the existence of non-measurable sets.

Hugh Woodin, The Continuum Hypothesis, Part II

This continues the previous article and discusses in what sense the continuum hypothesis can be considered true or false, even through it is formally independent of ZFC. MIDTERM in class on Monday , Oct 23 (tentative). Midterm 1 sample exam in PDF

FINAL is scheduled for Tuesday, Dec 13, 2:15-4:45

Chapter 1: Motivation and preliminaries

Problem set 1 is due Friday, Sept 16. Rewrites for full credit due Friday, Sept 23.
Problem set 1 in PDF
Problem set 1 in TeX

Wikipedia article on the Banach-Tarski paradox

Wikipedia article on Carleson's a.e. convergence theorem

Wikipedia article on Weierstrass' nowhere differentiable function

Chapter 2: Measure

Problem Set 2a due Monday, Sept 26
Problem set 2a in PDF
Problem set 2a in TeX

Problem Set 2b due Monday Oct 3
Problem set 2b in PDF
Problem set 2b in TeX

Chapter 3: Measureable functions
Problem Set 3 --- Due Friday Oct 14
Problem set 3 in PDF
Problem set 3 in TeX

Chapter 4: Integral ---
Problem Set 4
Problem set 4 in PDF
Problem set 4 in TeX

Chapter 5: Spaces of Integrable functions
Problem Set 5 --- Due
Problem set 5 in PDF
Problem set 5 in TeX

Chapter 6: Product measures
Problem Set 6 --- Due Mon, Nov 28
Problem set 6 in PDF
Problem set 6 in TeX

Chapter 7: The Radon-Nikodym theorem
No problem set for this chapter

The final will be 2:15-4:45pm in Rooom 4-130 of the math building (our usual classrooom). Sample final

Here are some `fun' problems to think about:
- Given a set X in the real numbers, how many different sets can you generate by taking complements and closures repeatedly?
-Show that every real number in the interval [0,2] Can be written as the sum of two real numbers in the Cantor middle thirds set.

Send me email at: bishop at

University final exam schedule

Link to Schroder-Bernstein theorem

Link to Freilng's dart argument against CH

Link to history of mathematics

Some specific topics from the history of math site: history of `e' , The Brachistochrome problem , Isaac Newton , Gottfried Willhelm von Leibniz , A brief history of calculus , The fundamental theorem of algebra , A brief history of mathematics , Jean Fourier , The number `Pi' , Discovery of Neptune and Pluto , ,