## Graduate Real Analysis I

Tuesday and Thursdays 11:10-12:25 in Boyd 410

Office Hours: TBA, during Math 8105 (see below), and by appointment
(Math 8105 meets with me at 1:50-2:40 on Wednesdays in Boyd 410)

Syllabus for 2021

Old Course Webpages from 2014 and 2018 and 2019

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 (discussion only, no proofs)
• 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 (in Math 8105)
Homework 1    (Due Thursday September 2)

• 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
• Lebesgue measure   (Section 1.3 in Stein)
• Measurable sets form a sigma-algebra
• Closed sets are measurable (only using that countable unions of measurable sets are measurable)
• Different characterizations of Lebesgue measurability (using closed sets, G-delta sets and F-sigma sets)
• Countable additivity and the briefest of introductions to abstract measure spaces
• Continuity from above and below
• Translation invariance
Homework 2    (Due Tuesday September 14)
• 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)
• Littlewood's Three Principles  (Section 1.4 in Stein)
• Every measurable set is nearly a "very nice set", such as a finite union of cubes or an open set
• The Lebesgue Density Theorem is arguably one of the strongest realizations of this principle [Statement only, proof is non-examinable]
• 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 Thursday September 23)

• 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 and Fatou's Lemma
• Integration of extended real-valued and complex-valued measurable functions  (Section 2.3 in Folland)
• Space of all complex-valued integrable functions form a complex vector space on which the Lebesgue integral is a complex linear functional
• Dominated Convergence Theorem
• Discussion on how to establish the continuity and differentiablity of functions defined by integrals
Homework 4    (Due Tuesday October 5)
• Definition of L^1, Completeness, and interchanging sums and integrals  (Section 2.3 in Folland continued, see also Section 2.4 in Folland and Section 2.2 in Stein)
• Examples of different modes of convergence
• Every sequence in L^1 which converges in norm contains a subsequence that converge almost everywhere
• Every absolutely convergent series in L^1 converges almost everywhere (and in L^1) to an L^1 function
• 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
• Further Properties of the Lebesgue Integral
• 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)
• An Approximation Theorem  (Section 2.2 in Stein, see also Section 2.3 and 2.6 in Folland)
• 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 14th of October)
Old Exams for practice: 2019 (75 minutes),  2018 (60 minutes),  2014 (60 minutes), and 2013 (75 minutes)

• Repeated Integration
Homework 5    (Due Tuesday October 26)
Homework 6    (Due Thursday November 4)

• Some Functional Analysis
Homework 7    (Due Monday the 22nd 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 examinable approach to the RRT for finite measure spaces when 1=<p<2.
• ** Discussion on the dual of L^∞ and the Hahn-Banach theorem [did not happen so non-examinable]
• The Riesz Representation Theorem (I will try to discuss this after Thanksgiving briefly)
Homework 8    (Due Thursday the 2nd of December)

* 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) [non-examinable]
• ** Complex measures (from Chapter 6 of Rudin's Real and Complex Analysis) [not covered in class, so non-examinable]
• Construction of Measures:
• Outer measures, metric outer measures and Caratheodory's theorem [non-examinable]
• 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) [non-examinable]
• Application to realizing the dual of C([0,1])
• Hausdorff measure and dimension

• More on Differentiation
•  The Lebesgue differentiation theorem [non-examinable]
• Motivation from the Fundamental Theorem of Calculus (Part I) and the Radon-Nikodym Theorem [non-examinable]
• The Hardy-Littlewood maximal function [not covered in class]
• Proof of the Lebesgue differentiation theorem [not covered in class]
• 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 [non-examinable]
• Monotone right-continuous functions are differentiable almost everywhere (using the associated Lebesgue-Stieltjes measure, Radon-Nikodym and Lebesgue differentiation theorems) [only briefly discussed with no proofs, non-examinable]
• Discussion of the notion of absolute continuity  [non-examinable]
• The Fundamental Theorem of Calculus (Part II) [non-examinable]