 ## Math 8100 - Real Analysis I - Fall 2019

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

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 in Stein)
• Preliminaries (decomposition theorems for open sets)
• Properties of Lebesgue outer measure
Homework 2    (Due Thursday September 5)

• Lebesgue measurable functions  (Section 1.4 in Stein)
• Equivalent definitions and the useful characterization using open sets
• Compositions and closure under algebraic and limiting operations
• Approximation by simple functions (and step functions) [Statement only in case of step functions, proof is non-examinable for now]
• Littlewood's Three Principles  (Section 1.4 in Stein)
• 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) [Statement only, proof is non-examinable for now]
• 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 2.2 in Folland)
• 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  (Section 2.3 in Folland)
• Provisional definition of L^1 and "the" Dominated Convergence Theorem
• Discussion on how to establish the continuity and differentiablity of functions defined by integrals
• Relationship with Riemann integration  (Theorem 2.28 in Folland)
• Absolute continuity of the Lebesgue integral and "small tails property"  (using the MCT as in proof of Proposition 1.12 in Section 2.1 of Stein)
• Translation and dilation invariance of the Lebesgue integral  (see  page 73 of Stein)
Homework 4    (Due Tuesday October 1)
• Completeness of L^1 and interchanging sums and integrals  (Section 2.3 in Folland continued, see also Section 2.4 in Folland and Section 2.2 in Stein)
• 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
• Every sequence in L^1 which converges in norm contains a subsequence that converge almost everywhere
• Conclusion of proof that L^1 is a complete normed space (a Banach space)
• Discussion that more generally a normed vector space X is complete if and only every absolutely convergent series in X converges
• An Approximation Theorem  (Section 2.2 in Stein, see also Section 2.3 and 2.6 in Folland)    [non-examinable for now]
• 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   (Proposition 2.5 in Section 2.2 of Stein)
• Short proofs of both the absolute continuity and "small tails property" of the Lebesgue integral

Exam 1
(In class on Thursday the 10th of October)
Here is Exam 1 from 2018 (60 mins),  Exam 1 from 2014 (60 mins), and Exam 1 from 2013 (75 mins) for practice

• Repeated Integration
Homework 5    (Due Friday October 18)
Homework 6    (Due Thursday October 31)

• Some Functional Analysis
Homework 7    (Due Thursday the 14th of November)

• Function Theory of L^p Spaces
• Linear Functionals and Dual spaces (for general normed vector spaces)
• The Dual Space of L^p (including a sketch proof of the Riesz Representation Theorem for L^p functions)     ** We gave a different approach to the RRT for p=1
• ** Discussion on the dual of L^∞ and the Hahn-Banach theorem [non-examinable]
• The Riesz Representation Theorem (statements only)
Homework 8    (Due Tuesday the 26th of November)

Exam 2
We will not have an in-class Exam 2. To help you study for the final, here is Exam 2 from 2018, Exam 2 from 2014 and a Practice Exam 2 (from 2014).

• Abstract Measure Theory
• Introduction
• Abstract measure spaces (definitions and examples)
• Integration on measure spaces
• Absolute continuity and the Radon-Nikodym theorem (von-Neumann's proof using the RRT for L^2 spaces)
• ** Complex measures (from Chapter 6 of Rudin's Real and Complex Analysis) [not covered in class]
• Construction of Measures:
• Outer measures, metric outer measures and Caratheodory's theorem
• Regularity of finite Borel measures
• (Caratheodory's) Extension theorem
• Product measures and Fubini/Tonelli
• Two Examples:
• Finite Borel measures and increasing right-continous functions (the Lebesgue-Stieltjes integral)
• Application to realizing the dual of C([0,1])
• Hausdorff measure and dimension

• More on Differentiation
•  The Lebesgue differentiation theorem
• Motivation from the Fundamental Theorem of Calculus (Part I) and the Radon-Nikodym Theorem
• The Hardy-Littlewood maximal function
• Proof of the Lebesgue differentiation theorem
• Pointwise convergence of approximate identities  [not covered in class]
• Rademacher functions, Random Series and the Strong Law of Large Numbers
• Returning to Lebesgue's Theorem on the Differentiation of a Monotone Function
• Monotone right-continuous functions are differentiable almost everywhere (using the associated Lebesgue-Stieltjes measure, Radon-Nikodym and Lebesgue differentiation theorems)
• Discussion of the notion of absolute continuity
• The Fundamental Theorem of Calculus (Part II)