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
  

• Introduction
• Recalling the Riemann integral and its limitations
• Can we assign a "measure" to all subsets of the real line?

• 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 (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
• Steinhaus' theorem (and non-measurable sets revisited)
• Lebesgue Density Theorem [Statement only, proof is non-examinable for now]
  

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

• Fubini-Tonelli Theorem(s)  (Section 2.3)
• Statement of theorems and discussion of examples
• Appendix on Measurability of "cylinder sets and functions"
  
• Proof of Fubini's theorem and Tonelli's theorem

Homework 5    (Due Friday October 18)

• Applications
  
• Lebesgue integral under linear transformations (example of an "it suffices to consider Borel measurable functions" argument) 
• Convolutions: 
  
• Basic Properties
• Approximations to the identity in L^1 (and in uniform norm for function which are bounded and uniformly continuous)
  
• Application of L^1 result: Smooth functions with compact support are dense in L^1
• Application of uniform norm result: Weierstrass approximation theorem
• Introduction to the Fourier transform on R^n and proof of the Fourier inversion formula
• Informal discussion on PDEs: Heat diffusion, Dirichlet problem for the upper-half plane, wave equation and the Cauchy problem

Homework 6    (Due Thursday October 31)

• Some Functional Analysis
  

• Hilbert Spaces
• Inner product spaces and Hilbert spaces
• Orthonormal sets and Fourier series
• Existence of a basis (for separable spaces this follows from Gram-Schmidt, in the non-separable case one requires the axiom of choice)
  
• Construction of an orthonormal basis for L^2(0,1) and the (periodic) Weierstrass approximation theorem on C([0,1])
• Handout on Fourier Series
  
• Closed subspaces, orthogonal projections and the Riesz Representation Theorem (for Hilbert Spaces)

Homework 7    (Due Thursday the 14th of November)

• Basic Theory of L^p Spaces
  
• Holder's inequailty, Minkowski's inequality and Completeness
  
• Dense subclasses
• ** Qualitative and quantitative relationships between the different L^p space (applications of Holder) [proofs not covered in class and non-examinable]

• 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
  
• "Converse of Holder"      ** We only discussed "Part (i)"
• ** 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    
• Supplement: Minkowski 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)

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