MAT 523 -- Real Variables I 
Course Information

 
Professor: D. Bradley

Office: 322 Neville Hall 
Hours: 11:00-11:50 a.m. MWF^* 
Phone: (207) 581-3920

Grades:

This course will operate somewhat like a seminar in integration theory. Despite the somewhat misleading title, the text provides an excellent introduction to the gauge integral of Kurzweil and Henstock, and the Lebesgue integral via McShane's Riemann-like adaptation of the Kurzweil-Henstock theory. Grades will be assigned on the basis of in-class participation and successful completion of written homework assignments.

Text:

Washek F. Pfeffer, The Riemann Approach to Integration: Local Geometric Theory, Cambridge Tracts in Mathematics #109, Cambridge University Press, 1993. ISBN: 0-521-05682-9.

Syllabus: I plan to cover most of chapters 1 through 6. In slightly more detail:

• Part I. One-dimensional integration
• Chapter 1. Preliminaries
• Lengths
• Partitions
• Stieljes sums
• Chapter 2. The McShane integral
• The integral
• Absolute integrability
• Convergence theorems
• Connections with derivatives
• Gap functions
• Integration by parts
• Chapter 3. Measure and measurability
• Extended real numbers
• Measures
• Measurable sets
• Calculating measures
• Negligible sets
• Measurable functions
• The a_A-measure
• Chapter 4. Integrable functions
• Integral and measure
• Semicontinuous functions
• The Perron test
• Approximations
• Chapter 5. Descriptive definition
• AC functions
• Covering theorems
• Differentiation
• Singular functions
• Chapter 6. The Henstock-Kurzweil integral
• The P-integral
• Integration by parts
• Connections with measures
• AC_*   functions
• Densities
• Almost differentiable functions
• Gages and calibers
• Additional topics as time permits. The extension to multidimensional integration in chapters 7 through 13 is planned for the sequel to this course, namely MAT 524, which will be offered in the spring semester of 2009.

Reference Texts:

• Robert G. Bartle, A Modern Theory of Integration, Graduate Texts Studies in Mathematics Volume 32, American Mathematical Society, 2000. ISBN: 0-8218-0845-1.

  I used this text when I taught this course in 2006--2007. A solutions manual is available; see Robert G. Bartle Solutions Manual to A Modern Theory of Integration, Graduate Studies in Mathematics, American Mathematical Society, 2000. ISBN: 0-8218-2821-5.

• Peng-Yee Lee and Rudolf Vyborny, The integral: An Easy Approach after Kurzweil and Henstock, Cambridge University Press, 2000.
• Russell A. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics Volume 4, American Mathematical Society, 1994.

  This is an advanced text explicating the various integration theories. The Lebesgue theory is used to prove properties of the Henstock integral rather than using McShane's approach to develop the Lebesgue integral via the Henstock-Kurzweil theory.

• Robert M. McLeod, The Generalized Riemann Integral, Carus Monograph Number 20, Mathematical Association of America, 1980.
• Charles Swartz, Introduction to Gauge Integrals, World Scientific, 2001.

  This is an undergraduate level introduction.

Reference Articles:

• Robert G. Bartle, Return to the Riemann integral, American Mathematical Monthly, vol. 103 (1996), no. 8, 625--632.
• Edward J. McShane, Unified integration, American Mathematical Monthly, vol. 80 (1973), no. 4, 349--359.
• Brian S. Thomson, Rethinking the elementary real analysis course, American Mathematical Monthly, vol. 114, no. 6 (June-July 2007), 469--490.

Websites:

• Drop the Riemann Integral Project http://classicalrealanalysis.com/drip.aspx
• Free Introduction to the Gauge Integral http://www.math.vanderbilt.edu/~schectex/ccc/gauge/