UT Dallas, Math 6301, Real Analysis
Fall 2024
John Zweck
Course Materials
Syllabus
Mathematics Professors and Mathematics Majors'
Expectations of eectures in Advanced Mathematics
[Keith Weber, AMS Blog]
Lecture Notes
0. Notes on the historical context for Lebesgue Measure and Integration with
scaffolded problems on pathological sets and functions
1. Point Set Topology Review
2. Continuous Functions Review
3. Course Overview
4. Riemann Integral Review
5. Why Do We Need Lebesgue?
6A. Construction of Lebesgue Measure: Polygons and Open Sets
6B. Construction of Lebesgue Measure: Polygons and Open Sets
7. Construction of Lebesgue Measure: Compact Sets
8. The Cantor Set
9A. Inner and Outer Measure
9B. Lebesgue Measure and its Properties
9C. Invariance of Lebesgue Measure
10. Sigma Algebras and Measurable Functions
11A. Pointwise, L-infinity convergence and Egorov's Theorem
11B. Convergence in measure
12. Lusin's Theorem
13. Lebesgue Integral: Nonnegative functions
14. Lebesgue Integral: General functions
15. Relationship between Riemann and Lebesgue Integrals
16. Examples and Applications of Convergence Theorems
17. Abstract Measure Spaces
18. Fubini's Theorem
Homework Assignments
Homework 1 [Due Thursday 29 Aug]
Homework 2 [Due Thursday 5 Sep]
Homework 3 [Due Tuesday 24 Sep]
Homework 4 [Due Thursday 26 Sep]
Homework 5 [Due Tuesday 15 Oct]
Homework 6 [Due Tuesday 22 Oct]
Homework 7 [Due Tuesday 5 Nov]
Some Extra Problems
Homework 8 [Due Tuesday 19 Nov]
Homework 9 [Due Tuesday 3 Dec]
More Extra Problems