Advanced Real Analysis 1

Math 564, 2023 Fall

Course material

• T. Tao's blog: courses 245A and 245B
• R. Bass, Real Analysis for Graduate Students [free download]
• G. Folland, Real Analysis [link] (not required)
• A. Tserunyan, The Cantor–Schöder–Bernstein theorem [pdf]

Assessment

• 30% homework + 30% midterm + 40% final.
• There will be 6 homework assignments (one every two weeks), to be submitted on Crowdmark.
• The midterm will be held in the week of Oct 16. The date of the final will be announced later.

Topics

1. Measures, their construction and properties
1. Polish spaces, σ-algebras and Borel sets, measurable spaces
2. Measures and premeasures
3. Constructions of Bernoulli(p) measures on 2^ℕ​​ and the Lebesgue measure on ℝ​​^d
4. Carathéodory's extension theorem: outer measures and two different proofs (by C. Carathéodory and T. Tao)
5. Measurable and non-measurable sets
6. Pocket tools: increasing unions/decreasing intersections, Borel–Cantelli lemmas, measure exhaustion and application: Sierpiński's theorem for atomless measures
7. Borel measures: their regularity (for metric spaces) and tightness (for Polish spaces), the 99% lemma for Bernoulli(p) and Lebesgue measures
8. Applications: ergodic group actions and non-measurability of transversals
9. Locally finite Borel measures on ℝ​​ and increasing right-continuous functions
2. Measurable functions and integration
10. Measurable functions, Luzin's theorem, push-forward measures, and random objects such as random graphs
11. Borel and measure isomorphism theorems (sketches of proofs)
12. Simple functions and their integration, approximation of measurable functions by simple ones
13. Integration of non-negative functions, monotone convergence theorem, and Fatou's lemma
14. Integration of real/complex valued functions and L¹, dominated convergence theorem, and the density of simple functions in L¹
15. Properties of integrable functions: σ-finiteness of the support, 99% boundedness, absolute continuity of B ↦​​ ∫_B f dμ
16. Convergence in measure and relations between different modes of convergence
17. Product measures and the Fubini–Tonelli theorem
3. Measure differentiation, density, and a.e. differentiable functions
19. Orthogonality and absolute continuity of measures, Jordan decomposition
20. Signed measures and Hahn decomposition (proof via measure exhaustion)
21. The Lebesgue-Radon-Nikodym theorem and Radon-Nikodym derivatives
22. The Lebesgue differentiation theorem for ℝ​​^d: proof via the Hardy–Littlewood maximal function and the Vitali covering lemma
23. The Lebesgue density theorem for ℝ​​^d, the Lebesgue differentiation theorem for all locally finite Borel measures on ℝ​​^d
24. Characterization of the distribution functions on ℝ​​ whose associated measures are absolutely continuous/orthogonal with respect to Lebesgue measure
25. Absolutely continuous functions and the fundamental theorem of calculus for increasing functions
26. Finite Borel signed measures on ℝ​​^d and right-continuous functions of bounded variation, the fundamental theorem of calculus for functions of bounded variation
4. L^p spaces
27. Norms and Banach spaces, L¹ is a Banach space
28. L^p norm and Minkowski's inequality
29. Hölder's inequality and relations between L^p spaces
30. L² as the Hilbert space (only statements, no proofs)
31. Bounded linear transformations and functionals, equivalence of continuity and Lipschitzness, operator norm
32. The dual of L^p for 1 ≤ p < ∞ (sketch of proof)