Midterm 1 (pdf) Midterm 1 (LaTeX file) Midterm 1 (Solutions)
The final exam is to be submitted through Canvas. The deadline is Sunday December 6, 11:59pm
Final Exam (pdf) Final Exam (LaTeX file)
https://pitt.zoom.us/j/91541847558
Instructor: Piotr Hajlasz
Office: 420 Thackeray Hall (I will be working from home).
Email: hajlasz@pitt.edu
Office hours: by appointment
Textbook:
P. Hajlasz, Measure Theory
Other recommended books:
W. Rudin Real & Complex Analysis.Third edition.
S. Axler. Measure, Integration & Real Analysis. Graduate Texts in Mathematics, Springer, 2020.
L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions. Revised edition. Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2015.
(Chapters 1,2 and Section 3.1).
P. Hajlasz Geometric Analysis
(Selected sections).
Brunn-Minkowski, isodiametric, isoperimentric inequalities
Convergence in measure, a.e., In L^1$, and the Vitali theorem
Doubling measures and the Vitali covering theorem
Lebesgue differentiation theorem
Functions of bounded variation
Absolutely continuous functions
Change of variables, area and coarea formulas
The Rademacher and the Stepanov theorem
Applications of the Rademacher theorem
Course Grade: Homework (30%) + Class participation (15%) + Attendance (15%) + Two take home exams (20% + 20%).
The group will work on the homework due to week n+2. Each homework set will have selected problem(s) that the leader of the group will present in the week of n+2. If the group will not be able to solve these problems that is okay as some of the problems might be difficult.
In addition to working on the homework, the group should meet and discuss the lectures of the week n and n+1. If the students find some of the lectures confusing or difficult, this will be a time to formulate questions for the class meetings. If there are any questions, the leader will write them in LaTeX and submit either through Canvas or by email (I have to determine what is more convenient) at the end of the week n+1. However, if there are no questions, there is no need to submit them. Selection of students to the group and selection of a leader will be random.
I know that this is an unusual procedure, but since we will all work in solitude, it is important to maintain healthy personal relations with other students.
HW#1 (pdf) HW#1 (LaTeX file) Friday August 28.
HW#2 (pdf) HW#2 (LaTeX file) Friday September 4.
HW#3 (pdf) HW#3 (LaTeX file) Friday September 18
HW#4 (pdf) HW#4 (LaTeX file) Friday September 25
HW#5 (pdf) HW#5 (LaTeX file) Friday October 2+epsilon
HW#6 (pdf) HW#6 (LaTeX file) Friday October 23.
HW#7 (pdf) HW#7 (LaTeX file) Friday November 18.
HW#8 (pdf) HW#8 (LaTeX file) This is the final exam.
Groups:
1 (leader), 19, 12, 6, 11, 5
9 (leader), 13, 22, 21, 23, 17
15 (leader), 2, 7, 10, 4, 20
18 (leader), 8, 16, 3, 14
August 19 Lecture 1a (Sigma algebras and Borel sets.) Lecture 1b (Basic properties of measures.)
August 21 Lecture 2 (Outer measures and the Caratheodory theorem.)
August 24 Lecture 3a (Complete measures.) Lecture 3b (Metric outer measures.)
August 26 Lecture 4 (Regularity of measures and Radon measures.)
August 28 Lecture 5 (Hausdorff measures, Hausdorff dimension.)
August 31 Lecture 6a (Hausdorff dimension: Cantor set and van Koch curve) Lecture 6b (Hausdorff measure H^1 in R.)
September 2 Lecture 7 (Further properties of the Hausdorff measure.)
September 4 Lecture 8a (Lebesgue measure I.) Lecture 8b (Lebesgue measure II)
September 7 Lecture 9a (Characterization of Lebesgue measurable sests.) Lecture 9b (Not every set is measurable: Vitali's constuction.)
September 9 Lecture 10a (Uniqueness of translation invariant measures.) Lecture 10b (Sets of measure zero, Lipschitz mappings, Borel sets and homeomorphisms.)
September 11 Lecture 11 (Change of the Lebesgue measure under linear mappings.)
September 14 Lecture 12 (Measurable mappings. Borel mappings.)
September 16 Lecture 13 (Measurable functions; measurability of limsup, liminf, lim etc.)
September 18 Lecture 14 (Lebesgue integral, Lebesgue monotone convergence theorem.)
September 21 Lecture 15 (Fatou's lemma and the Dominated Convergence Theorem.)
September 23 Lecture 16 (Almost everywhere. Identification of funcitons equal a.e. Existence of a Borel representative.)
September 25 Lecture 17a (The Lusin theorem.) Lecture 17b (The Egorov theorem.)
September 28 Lecture 18 (Theorems of Lebesgue and Riesz: convergence in measure and convergence a.e. Absolute continuity of the integral. Uniform integrability. The Vitali theorem - generalization of the dominated convergence theorem.)
September 30 Lecture 19a (Applications of the Brunn-Minkowski inequality: isodiateric inequality and equality between teh Lebesgue and the Hausdorff measures) Lecture 19b (Isoperimetric inequality and the proof of the Brunn-Minkowski inequality.)
October 2 Lecture 20 (Fubini theorem.)
October 5 Lecture 21a (L^p spaces, definition) Lecture 21b (Jensen's inequality) Lecture 21c (Holder's inequality)
October 7 Lecture 22 (L^p as a Banach space)
October 9-16 Lecture 23a (Convolution in L^1 and in L^p) Lecture 23b (Approximation by convolution. Part 1) Lecture 23c (Approximation by convolution. Part 2)
October 19-23 Lecture 24a (Doubling measures) Lecture 24b (5r covering lemma) Lecture 24c (Vitali covering theorem)
October 26 Lecture 25 (Doubling measures, s-regular measures, and Hausdorff measures)
October 28-30 Lecture 26a (Lebesgue differentiation theorem I) Lecture 26b (Lebesgue differentiation theorem II)
November 2 Lecture 27 (Maximal function and the Lebesgue differentiation theorem)
November 4 Lecture 28 (Functins of bounded variation)
November 6 Lecture 29 (Absolutely continuous functions)
November 9 Lecture 30 (The Rademacher and the Stepanov theorems)
Recitations activities
Fubini and and an application of convolution