Sept 11
Course introduction, algebras, σ-algebras
Folland Chapter 1.1–2
Sept 18
Borel σ-algebra, Definition and basic properties of measures
Folland Chapter 1.3
Sept 20
Null sets and completion, outer measures, Carathéodory's
Theorem, extension
Folland Chapter 1.3–4
Sept 25
Premeasures on an algebra and extension
Folland Chapter 1.4
Sept 27
Construction of the Lebesgue-Stieltjes measure; properties
Folland Chapter 1.5
Oct 4
Measurable functions, characteristic functions, simple
functions and approximation
Folland Chapter 2.1
Oct 11
The rôle of null sets, Integration of simple functions and
non-negative functions. Monotone Convergence Theorem and Fatou's Lemma.
Folland Chapter 2.2
Oct 16
Integral of complex-valued functions, L¹ as a vector space,
Dominated Convergence Theorem
Folland Chapter 2.3
Oct 18
Proof of the Dominated Convergence Theorem, Normed vector
spaces, completeness criterion, Banach spaces.
Folland Chapter 2.3, 5.1
Oct 23
Lp spaces, Hölder inequality and Minkowski
inequality.
Folland Chapter 2.3, 6.1
Oct 25
L∞ space, ℓp spaces, interpolation
inequalities. Bounded linear operator and functionals on Banach spaces
Folland Chapter 6.1, 5.1
Oct 30
Midterm review session
Nov 1
Midterm exam (room EX200)
Nov 13
Dual of Lp
Folland Chap. 6.2
Nov 15
Lebesgue vs Riemann Integral, modes of convergence, Product
measures
Folland Chap 2.3–5
Nov 20
Signed measures, Hahn and Jordan Decomposition, absolute
continuity
Folland Chap 3.1–2
Nov 22
The Radon-Nikodym-Lebesgue Theorem
Folland Chap 3.2-3.3
Nov 27
Complex measures, the dual of Lp
Folland Chap 3.3 6.2
Nov 29
Differentiation on Euclidian space
Folland Chap 3.4