Math 8100 - Real Analysis I - Fall 2019

Graduate Real Analysis I

 Tuesday and Thursdays 11:00-12:15 in Boyd 303

Office Hours: MF 1:25-2:15, during Math 8105 (see below), and by appointment
(Math 8105 meets 1:25-2:15 in  Boyd 410)

 
Syllabus for 2019

Old Course Webpages from 2014 and 2018

Principal Textbook:
Real Analysis, by E. M. Stein and R. Shakarchi

Secondary References:

Real Analysis, by G. B. Folland

An introduction to measure theory, by Terrence Tao  
Real and Complex Analysis, by W. Rudin


Course Outline (eventually including Assignments) and Some Supplementary Class Notes

For the most part we shall follow closely the appropriate sections of the reference(s) listed above.

  • Preliminaries
    • Three Notions of Smallness for subsets of the Reals
      • Countable, Meager (1st Category), and Null (measure zero)
      • The Reals are "not small", specifically they are neither countable, meager, or null (Theorems of Cantor, Baire, and Borel respectively)
    • Discontinuities and Non-Differentiability
      • F-sigma sets, first class functions, and Lebesgue's Criterion for Riemann Integrability
      • Nowhere differentiable functions and Lebesgue's Theorem on the differentiablity of monotone functions
    • Review of Uniform Convergence and Uniform Continuity
Homework 1    (Due Tuesday August 27)

  • Lebesgue Measurable Sets and Functions
    • Lebesgue outer measure   (Sections 1.1 and 1.2)
      • Preliminaries (decomposition theorems for open sets)
      • Properties of Lebesgue outer measure
Homework 2    (Due Thursday September 5)

    • Lebesgue measurable functions  (Section 1.4)
      • Equivalent definitions and the useful characterization using open sets
      • Compositions and closure under algebraic and limiting operations
      • Approximation by simple functions (and step functions)
    • Littlewood's Three Principles  (Section 1.4)
      • Every measurable set is nearly an open set (the Lebesgue Density Theorem is arguably the strongest realization of this principle)
      • Every measurable function is nearly a continuous function (Lusin's theorem)
      • Every convergent sequence of measurable functions is nearly uniformly convergent (Egorov's theorem)
Homework 3    (Due Friday September 20)

  • Lebesgue Integral
    • Integration of non-negative measurable functions  (Section 2.1)
      • Simple functions and their integral (including properties such as monotonicity and linearity)
      • Extension to all non-negative measurable functions (establishing linearity using MCT below)
        • Monotone Convergence Theorem (proved using "continuity from below" for measures defined by simple functions)
        • Chebyshev's inequality and proof that "f equals 0 a.e. if and only if the integral of f equals 0"
      • Convergence Theorems for non-negative measurable functions and examples
        • (Modified) Monotone Convergence Theorem, Fatou's Lemma, and the Dominated Convergence Theorem
        • Discussion that these results are "equivalent"
    • Integration of extended real-valued and complex-valued measurable functions
      • Provisional definition of L^1 and "the" Dominated Convergence Theorem
        • Discussion on how to establish the continuity and differentiablity of functions defined by integrals
      • Absolute continuity of the Lebesgue integral and "small tails property" (using the Monotone Convergence Theorem)
      • Translation and dilation invariance of the Lebesgue integral
      • Relationship with Riemann integration

Homework 4    (Due Tuesday October 1)

    • Completeness of L^1 and interchanging sums and integrals
      • Every absolutely convergent series in L^1 converges almost everywhere (and in L^1) to an L^1 function
      • Examples of different modes of convergence
        • Sequences in L^1 which converge in norm contain subsequences that converge almost everywhere
      • Conclusion of proof that L^1 is a complete normed space (a Banach space)
    • An Approximation Theorem   
      • Simple functions and Continuous functions with compact support are both dense in L^1 -- another realization of Littlewood's second principle
        • Continuity in L^1
        • Short proofs of both the absolute continuity and "small tails property" of the Lebesgue integral