I am teaching MA947 - Graduate real Analysis during term I. All the information can be found at the university webpage.
The course will follow the lecture notes written by David Bate that you can find either on the moodle or here.
The course will be organized as follows:
Chapter 1: Measures. (Lectures 1, 2, 3, 4).
Chapter 2: Integration (Lectures 5, 6).
Chapter 3: Egorov's theorem, Lusin's theorem, Radon-Nikodym theorem, Lebesgue decomposition theorem (Lectures 7, 8, 9).
Chapter 4: Riesz's representation theorem, lattices of functions and Daniell integral (Lectures 10, 11, 12, 13, 14).
Chapter 5: Fubini's theorem. (Lectures 15, 16, 17).
Chapter 6: Covering theorems, Hardy-Littlewood maximal function and Lebesgue differentiation (Lectures 18, 19, 20, 21).
Chapter 7: Lipschitz functions. AC and BV in the real line. Rademacher Theorem (Lectures 22, 23, 24, 25).
Chapter 9: Rectifiable sets and approximate tangent planes (Lectures 26, 27, 28, 29).
Chapter 10: Purely unrectifiable sets (Lecture 30).
List of typos in the lecture notes:
Exercise 3.5. \nu needs be replaced by \mu on the right hand side of the equation in display.
Pag. 15. There is a typo in the definition of monotonicity of T. It should be T(f) \le T(g).
Pag. 15. There is a typo in the definition of `T is continuous with respect to monotone convergence'. We need to assume that f_i and f belong to the lattice L.
Pag. 15, line 12, 13. A and B need to be replaced by E \cap A and E \cap B.
Pag. 22, 8th line after (5.2): V should be replaced by U.
Pag. 29: second equation. It is not a typo, just a reminder of what is the meaning of the notation. It means that for every A \subset X, \mu(A)=\int_A \frac{d \mu}{d \mathcal{L}^1} d \mathcal{L}^1.
Some interesting material about real analysis and geometric measure theory useful for the course is:
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I was a Teaching assistant for the course of Complex Analysis II (held by Behnam Esmayli) in Spring 2023.
This is the solution to the exercise sheets we provided during the first week: Exercise sheet (solved).Â
For all the other weeks, we will solve the exercises given by Behnam during classes, that you may find here.