## Math 8100 - Real Analysis I - Fall 2018

Monday, Wednesday, and Fridays 11:15-12:05 in Boyd 302

Office Hours: Mondays and Wednesdays 12:15-1:15, during Math 8105 (see below), and by appointment
(Math 8105 meets from 2:00-3:15 on Thursdays in Boyd 222)

Syllabus for 2018

Old Course Webpage from 2014

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 Friday the 24th of August)

• 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 Wednesday the 5th of September)

• 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 Wednesday the 19th of September)

• 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 Friday the 28th of September)

• 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

Exam 1
(In class on Wednesday the 10th of October)
Here is Exam 1 from 2014 and Exam 1 from 2013 for practice, but please note that this was a 75 minute exam in 2013

• Repeated Integration
Homework 5    (Due Friday the 19th of October)
Homework 6    (Due Wednesday the 31st of October)

• Some Functional Analysis

Homework 7    (Due Friday the 9th 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 this tear in some special cases
• ** Discussion on the dual of L^∞ and the Hahn-Banach theorem [non-examinable]
• The Riesz Representation Theorem (statements only)
Homework 8    (Due Friday the 16th of November)

Exam 2   (In class on Wednesday the 28th of November)
Here is Exam 2 from 2014 and a Practice Exam 2 (from 2014) which is longer than an actual exam would be, but should hopefully still help with your studying!

• 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]
• 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)