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.

• 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)
  
• Preliminaries (decomposition theorems for open sets)
• Properties of Lebesgue outer measure

• Lebesgue measure I   (Section 1.3)
  
• Measurable sets form a sigma-algebra
• Closed sets are measurable (only using that countable unions of measurable sets are measurable) 

Homework 1    (Due Wednesday the 27th of August)

• Lebesgue measure II   (Section 1.3)
• Countable additivity and the briefest of introductions to abstract measure spaces (including continuity from above and below)
  
• Translation invariance
• Steinhaus' theorem (and non-measurable sets revisited)
• Different characterizations (using closed sets, G-delta sets and F-sigma sets)

Homework 2    (Due Friday 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

• Littlewood's Three Principles  (Section 1.4)
• Every measurable set is nearly an open set
• Every measurable function is nearly a continuous function (Lusin's theorem)
• Every convergent sequence of measurable functions is nearly uniformly convergent (Egoroff's theorem)

Homework 3    (Due Friday the 12th of September)

• Lebesgue Integral

• Lebesgue Integral I: 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)

• Lebesgue Integral II: The Convergence Theorems  (Section 2.1)
• Non-negative measurable functions:
• 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"
• Modified Monotone Convergence Theorem and Fatou's Lemma (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

Homework 4    (Due Friday the 19th of September)

• Lebesgue Integral III: Applications of the Convergence Theorems  (Section 2.1)
• Sums and integrals: Every absolutely convergent series in L^1 converges almost everywhere (and in L^1) to an L^1 function
• Absolute continuity of the Lebesgue integral and "small tails"
• Translation and dilation invariance
• Relationship with Riemann integrable functions
  

• Lebesgue Integral IV: The normed vector space L^1  (Section 2.2)
• Simple functions and continuous functions with compact support are dense in L^1
  
• Continuity in L^1
• Modes of Convergence: Discussion of examples
• Proof that L^1 is a complete normed space (a Banach space)
• Sequences which are convergent in norm have subsequences that converge almost everywhere

Homework 5    (Due Friday the 26th of September)

Exam 1   (In class on Monday the 6th of October)

Here is Exam 1 from 2013 for practice, but please note that this was a 75 minute exam

• Repeated Integration

• Fubini-Tonelli Theorem(s)  (Section 2.3)
• Statement of theorems and discussion of examples
• Proof of Fubini's theorem and Tonelli's theorem

Homework 6    (Due Wednesday the 15th of October)

• 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 [non-examinable]

Homework 7    (Due Friday the 24th of October)

• Hilbert Spaces (and a little Fourier series)
  

• 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])
• Closed subspaces, orthogonal projections and the Riesz Representation Theorem (for Hilbert Spaces)

• ** More on the pointwise convergence of Fourier Series [not covered in class and non-examinable]:
• Consequences of convergence in L^2 to the question of pointwise convergence
• Dini's Criterion for Pointwise Convergence
• Cesaro Means and Fejer's Theorem
• Poisson Summation Formula

Homework 8    (Due Friday the 7th of November)

• L^p Spaces
  

• 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)
• "Converse of Holder"

• ** Additional L^p theory [not covered in class and non-examinable]:
  
• "Continuity in L^p" and Minkowski's integral inequality
• A special case of Young's inequality and approximations to the identity in L^p (applications of Minkowski's inequality for integrals)
• Discussion of Young's inequality

Homework 9    (Due Friday the 14th of November)

• Abstract Measure Theory
  

• Introduction
  
• Abstract measure spaces (definitions and examples)
• Integration on measure spaces
• Complex Measures, absolute continuity and the Radon-Nikodym theorem
• ** More on complex measures (from Chapter 6 of Rudin's Real and Complex Analysis) [proofs not covered in class and non-examinable]

Homework 10    (Due Friday the 21st of November)

• Representation Theorems
• Riesz Representation Theorem for Hilbert Spaces (and in particular for L^2 spaces)
  
• Radon-Nikodym theorem (von-Neumann's proof using the RRT for L^2 spaces)
  
• Riesz Representation Theorem for L^p functions (proved only for finite measure spaces)
  
• ** Discussion on the dual of L^∞ and the Hahn-Banach theorem [non-examinable]
  
• The Riesz Representation Theorem (statements only)

Exam 2   (In class on Wednesday the 3rd of December)

Here is Exam 2 from 2013 for practice, but please note that this was a 75 minute exam (and the material was slightly different)

Here is a Practice Exam 2, this is longer than an actual exam would be, but should hopefully still help with your studying!

• ** Construction of Measures [non-examinable]:
• Outer measures, metric outer measures and Caratheodory's theorem
• Regularity of finite Borel measures
  
• (Caratheodory's) Extension theorem [not covered in class]
• Product measures and Fubini/Tonelli [not covered in class]
• 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

Homework 11    (Due Wednesday the 10th of December)

• More on Differentiation [not covered in class and non-examinable]

• ** Introduction

• ** The Lebesgue differentiation theorem
• The Hardy-Littlewood maximal function
• Proof of the Lebesgue differentiation theorem
• Pointwise convergence of approximate identities
  

• ** Discussion of the notion of absolute continuity
• The Fundamental Theorem of Calculus (Part II)
  

Final Exam