12/6 Friday
We went over Homework 12 problems.
12/3 Tuesday
We proved the Fubini's Theorem.
11/27 Wednesday
Homework 12 is posted.
11/26 Tuesday
We proved the Riesz-Fischer theorem that L^1 is a complete with respect to L^1 norm. After Thanksgiving break, we will prove Fubini's theorem.
11/22 Friday
We started Section Section 2.2 The Space of Integrable Functions and discussed L^{1} space.
11/19 Tuesday
We proved that Riemann integrable functions are measurable, Lebesgue integrable, and two integrals coincide. On Friday, we will cover Section 2.2 The Space of Integrable Functions.
11/15 Friday
We proved the Monotone Convergence Theorem and Lebesgue Diminated Convergence Theorem.
11/12 Tuesday
We started proving limit theorems; bounded convergence theorem, Fatou's lemma, monotone convergence theorem, and Lebesgue dominated convergence theorem. We proved bounded convergence theorem and Fatou's lemma in class and we will prove the other two on Friday.
11/7 Friday
We discussed Section 2.1 The Lebesgue integral: basic properties and convergence theorems. We covered up to page 60 except the part on Return to Riemann integrable functions, which will be covered next week. Next week, we will start proving important limit theorems such as Faou's lemma, monotone convergence theorem, and Lebesgue dominated convergence theorem.
11/5 Tuesday
We defined the Lebesgue integral for bounded functions supported on a set of finite measure using the Ergorov Theorem. On Friday, we will generalize the class of functions into all nonnegative measurable functions then arbitrary measurable functions.
11/1 Friday
We started Chapter 2 Lebesgue Integral and covered Lebesgue integral of simple functions. Next, we will define the Lebesgue integral for bounded functions with compact supports and for this we will need Ergorov Theorem.
10/29 Tuesday
We went over Midterm 2.
For the class on Friday, read (and try to understand) the Egorov Theorem in page 33.
10/25 Friday
We took Midterm 2. Homework 8 is posted.
10/22 Tuesday
We covered Section 4.2 Approximation by Simple and Step Functions. We will take midterm 2 on Friday 10/25.
10/18 Friday
We continued Section 1.4 Measurable Functions, and proved the the class of Lebesgue measurable functions is close under ordinary operations and taking limits.
10/11 Friday
We discussed Section 1.4 Measurable Functions.
10/8 Tuesday
We went over some questions from Homework 6. On Friday, we will start Section 1.4 Measurable Functions.
10/4 Friday
We constructed a non-measurable set and demonstrated an example that the additivity fails for the exterior measure.
10/1 Tuesday
We went over questions on midterm 1.
9/27 Friday
We took midterm 1.
9/24 Tuesday
We continue our discussion on properties of Lebesgue measurable sets. We introduced σ-algebra and the Borel σ-algebra. Homework 4 is due this Friday, and we will take Midterm 1 on Friday.
9/20 Friday
We continue our discussion on properties of Lebesgue measurable sets. Midterm 1 will be next Friday 10/27 in class and the midterm 1 will cover materials up to today.
9/17 Tuesday
We started Section 1.4 Measurable Sets and Lebesgue Measure and discussed properties of (Lebesgue) measurable sets.
9/13 Friday
We continued discussing properties of external measure. Homework 3 is posted. Homework 2 solution is posted.
9/10 Tuesday
We discussed properties of the exterior measure; monotonicity and countable sub-additivity. Homework 2 is due this Friday 9/13.
9/6 Friday
We construced the ternary Cantor set. Then, we introduced the exterior measure and started proving properties of it.
9/3 Tuesday
We discussed the volume of rectangles and structure of open sets in R and R^d.
Homework 1 is due on 9/6 Friday before the class starts.
8/30 Friday
We covered Section 1.1 Preliminaries. We introduced various definitions such as open/closed sets, boundary points, limit points, and compact sets, and Heine-Borel finite open covering property for compacts sets. Next week, we will work on volumes of rectangles and use this to define exterior measures for arbitrary sets in R^d.
8/27 Tuesday
We went over our syllabus and started a lecture on the complete axioms of the set of real numbers. Homework 1 is due 9/6 Friday.
8/26 Welcome to MAT 431 Real Analysis.
Homework 12 due 12/6 Friday, Solution
Homework 11 due 11/22 Friday, Solution
Homework 10 due 11/15 Friday, Solution
Homework 9 due 11/8 Friday, Solution
Homework 8 due 11/1 Friday, Solution
Homework 7 due 10/18 Friday, Solution
Homework 6 due 10/11 Friday, Solution
Homework 5 due 10/4 Friday, Solution
Homework 4 due 9/27 Friday, Solution
Homework 3 due 9/20 Friday, Solution
Homework 2 due 9/13 Friday, Solution
Homework 1 due 9/6 Friday by the start of the class