MAT 524 -- Functions of a Real Variable II 
Course Description

Professor: D. Bradley (http://www.umemat.maine.edu/~bradley/)

This course is a continuation of MAT 523. As such, we will be reinforcing many of the concepts and topics of MAT 523 as well as forging ahead with new material in the study of real analysis, integration, measure theory and functional analysis. One particularly exciting and relatively new development is the discovery of an integral more powerful than either the Riemann or Lebesgue integrals, but whose definition is as elementary as Riemann's, so that the need for the sophisticated machinery of measure theory is completely bypassed. This integral, (variously known as the Riemann-complete integral, gauge integral, Henstock-Kurzweil integral, Denjoy-Perron integral) integrates more functions than either Riemann's or Lebesgue's, including all derivatives, so it satisfies a much simpler version of the Fundamental Theorem of Calculus. Both the Riemann and Lebesgue integrals can directly handle only functions whose absolute values are integrable. So in one sense, this ''super integral" can be viewed as a continuous analog of summing conditionally convergent series, whereas both the Riemann and Lebesgue theories are restricted to the analog of summing only absolutely convergent series.

Students who will profit:

Anyone contemplating the pursuit of an advanced degree in a subject with technical requirements in higher mathematics. Most of the theory we now regard as part of modern "real analysis" had its origin in problems arising in applied mathematics, probability, engineering, and theoretical physics.

Text:

Brian S. Thomson, Theory of the Integral, ClassicalRealAnalysis.com (2013) ISBN: 978-1467924399. Free pdf copy downloadable at http://classicalrealanalysis.info/Theory-of-the-Integral.php

Topics may include:

• Extensions of Lebesgue Measure
• the continuum hypothesis, inaccessible cardinals, and the existence of non-trivial measures
• the Banach-Tarski paradox
• the role of the axiom of choice in the construction of non-measurable sets and discontinuous linear functions
• a model of set theory in which every set of reals is Lebesgue measurable
 
• Signed Measures, Complex Measures
• The Hahn/Jordan Decomposition and Radon-Nikodym Theorems
• Functions of Bounded Variation and Absolutely Continuous Functions
• The Fundamental Theorem of Calculus for Lebesgue Integrals
 
• Other Integration Theories
• the Daniell integral
• the Riemann-complete (a.k.a. gauge, Henstock-Kurzweil, Denjoy-Perron) integral
• non-absolutely integable functions
 
• Elements of Functional Analysis
  • Normed Vector Spaces, Linear Functionals
  • The Hahn-Banach Theorem
  • The Baire Category Theorem
  • Closed Graph and Open Mapping Theorems; Uniform Boundedness Principle
  • Hilbert Spaces
   
• L^p Spaces
• Hölder and Minkowski Inequalities
• Riesz-Fischer Theorem: Completeness of L^p
• Isometries of L^p, the dual of L^p
• the Riesz Representation Theorem for linear functionals on L^p
 
• Elements of Fourier Analysis
• Convolutions
• The Fourier Transform
• The Riemann-Lebesgue Lemma
• Fourier Inversion
• Plancherel's Theorem
• The Poisson Summation Formula
• Pointwise Convergence of Fourier Series
• Applications to Partial Differential Equations
 
• Introduction to Probability
• Independence
• Bernstein's proof of the Weierstrass approximation theorem
• Law of Large Numbers
• Central Limit Theorem
 

References:

• Robert G. Bartle, A Modern Theory of Integration, Graduate Studies in Mathematics Vol. 32, American Mathematical Society, 2001. ISBN: 0-8218-0845-1.
• Keith Devlin, The Joy of Sets: Fundamentals of Contemporary Set Theory, (2nd ed.) Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1993.
• Gerald B. Folland, Real Analysis: Modern Techniques and their Applications, (2nd ed.) John Wiley and Sons, New York, 1999.  Chapters 3, 5, 6, and others as time permits.
• 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.

• Paul R. Halmos, Measure Theory, Graduate Texts in Mathematics Vol. 18, Springer-Verlag, New York, 1950.  Chapters VI, IX & X.
• Peng-Yee Lee and Rudolf Vyborny, The Integral: An Easy Approach after Kurzweil and Henstock, Cambridge University Press, 2000.
• Robert M. McLeod, The Generalized Riemann Integral, Carus Monograph Number 20, Mathematical Association of America, 1980.
• John C. Oxtoby, Measure and Category, (2nd ed.) Graduate Texts in Mathematics Vol. 2, Springer-Verlag, New York, 1980.
• 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.
• Inder K. Rana, An Introduction to Measure and Integration, (2nd ed.) Graduate Texts in Mathematics Vol. 45, American Mathematical Society, 2002.
• M. M. Rao, Measure Theory and Integration, (2nd ed.) Marcel Dekker, New York, 2004.
• Halsey L. Royden, Real Analysis, (3rd ed.) MacMillan, New York, 1988.   Chapters 7-10, 13 & 16.
• Walter Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1987.  Chapters 2, 5-7 & 9.
• 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.info/resources-drip.php
• Free Introduction to the Gauge Integral http://www.math.vanderbilt.edu/~schectex/ccc/gauge/