Math 711 Homework

Homework Assignments

12/21
  1. Here are solutions to the take-home final.
  2. Have a good break!
12/03
  1. Finish reading section 3.5 (Functions of bounded variation) in Folland. Send me a question by Monday afternoon. Due 12/07.
  2. Exercises 3.5: Exercises 36 through 42.
  3. Here is the final exam. As with the midterm, you may discuss the questions amongst yourselves (but not with any students who may have taken this course in previous years), but you must write up your own paper to be handed in, and you must properly acknowledge any assistance you received from your peers. Due 12/18 (Noon).
12/01
  1. Read section 3.5 (Functions of bounded variation) in Folland. (There is a LOT of important information in this section, so get as far as you can, but don’t feel bad if you don’t finish it completely.)
  2. Exercises 3.5: Exercises 27 through 35.
11/17     I will be out of town for the next week. Please meet at our regular class time and work on the following material from the beginning of Chapter 5 of Folland. Reading should be done individually, as usual; you should try to answer each other’s questions during class or, alternatively, compile a list of class questions to send me by e-mail.

  1. Read Sections 5.1 (Normed vector spaces) and 5.2 (Linear functionals). Send me a question by next Monday evening. Section 5.1 due 11/19; Section 5.2 due 11/24.
  2. Exercises 5.1: Exercises (1), 6, 8, (9, 10, 11), 12, 13, (16). (Exercises are in parentheses are interesting but optional.)
  3. Exercises 5.2: Exercises 17, 18, 19, 20, (22, 23, 24, 26).
  4. Have a good Thanksgiving!
11/12
  1. Read section 3.4 (Differentiation in Euclidean space) in Folland. Send me a question by Monday evening. (This is a long section, and we may need to finish it next time, but read as far as you can.) Due 11/16.
  2. Hand in Exercises 3.3: 21; Exercises 3.4: 24, 25. Due 11/19.
11/10
  1. Read section 3.3 (Complex measures) in Folland, through the proof of Theorem 3.17.
  2. Exercises 3.2: Exercises 12, 15, 17.
  3. Exercises 3.3: Exercises 18, 19, 21.
11/05
  1. Read section 3.2 (The Lebesgue-Radon-Nikodym Theorem) in Folland and send me a question by Monday evening. Due 11/09.
  2. Exercises 3.2: Exercises 8, 9, 10.
  3. Hand in Exercises 3.1: 2, 4, 6; Exercises 3.2: 8. Due 11/12.
11/03
  1. Read section 3.1 (Signed measures) in Folland.
  2. Exercises 3.1: Exercises 1 through 7.
10/29
  1. Finish reading section 2.6 (The n-dimensional Lebesgue integral) in Folland, and read section 2.7 (Integration in polar coordinates)
  2. Exercises 2.6: 53, 55, 56, 58, 59.
  3. Exercises 2.7: 62, 63, 64.
10/27
  1. Read section 2.6 (The n-dimensional Lebesgue integral) in Folland, through Corollary 2.46.
  2. Exercises 2.5: 51, 52.
10/22
  1. Here are solutions to the midterm exam.
  2. Finish reading section 2.5 (Product measures) in Folland, and read the portion of Section 1.5 (Borel measures on the real line) that we omitted earlier (pages 35 through 39). Send me a question by Monday evening. Due 10/26.
  3. Exercises 2.5: 45, 48, 49, 50.
  4. Hand in Exercises 2.3: 22, 28(a); Exercises 2.4: 44. Due 10/29.
10/20
  1. Continue (or start) reading section 2.5 (Product measures) in Folland. Read at least as far as the proof of the Monotone Class Lemma. (We will continue reading from Theorem 2.36 next time.)
10/15
  1. Start reading section 2.5 (Product measures) in Folland. Read at least as far as the proof of the Monotone Class Lemma. (We will continue reading from Theorem 2.36 next time.)
10/13
  1. (Re)read section 2.4 (Modes of convergence) in Folland in light of today’s discussion. Send me a question by Wednesday evening. Due 10/14.
  2. Complete the midterm exam and hand it in next Tuesday. Due 10/20.
  3. Exercises 2.3: 19, 20, 21, 22, 28, 29; Exercises 2.4: 32, 35, 36, 38, 40, 44. (But finish the midterm first!)
10/01
  1. Read Section 2.3 (Integration of complex functions) in Folland. You should make sure you understand the statement of Theorem 2.28, but you can omit the proof if you wish: it will have become unnecessary by the end of the semester.
  2. Start working on the midterm exam. You may discuss the questions amongst yourselves (but not with any students who may have taken this course in previous years), but you must write up your own paper to be handed in, and you must properly acknowledge any assistance you received from your peers. Due 10/20.
09/29
  1. Re-read section 2.1 (Measurable functions) in Folland in light of today’s discussion.
  2. Complete the “Measurable Functions” class exercises set.
  3. Exercises 2.1: 1, 2, 3, 5. Exercises 2.2: 12, 13, 14.
  4. Read Section 2.2 (Integration of nonnegative functions) in Folland.
09/24
  1. Read section 2.1 (Measurable functions) in Folland. Send me a question on the reading by Monday afternoon. Due 09/28.
  2. Hand in Exercises 1.5: 33; Exercises 2.1: 8. Due 10/01.
  3. You will be asked to read Section 2.2 (Integration of nonnegative functions) for next Thursday. This is a longer section, so you might want to start reading ahead this weekend.
09/22
  1. Re-read section 1.5 in Folland in light of today’s discussion.
  2. Exercises 1.5: 27, 28, 32, 33.
  3. We will start reading Chapter 2 of Folland on Thursday. Feel free to read ahead now, if you wish.
09/17
  1. Re-read section 1.4 in Folland in light of today’s discussion, trying to put together both the big picture and the small details. In particular, see how the proof of Theorem 1.11 (Caratheodory’s Theorem) divides into the following steps:
    • M is closed under complements (where M denotes the σ-algebra of μ*-measurable sets);
    • M is an algebra;
    • μ* is finitely additive on M;
    • M is a σ-algebra;
    • μ* is countably additive on M;
    • μ* restricted to M is a complete measure.
  2. Read section 1.5 in Folland (Borel measures on the real line). You need only read the construction of Lebesgue-Stieltjes measures (which finishes about 2/3 of the way down page 35) and the discussion of the Cantor set (which starts with the last paragraph on page 37). We will come back to regularity properties of Lebesgue measure later.
  3. You may find this Wikipedia page on the Cantor function useful.
  4. Exercises 1.4: 17, 18, 19, 23.
  5. Hand in Exercises 1.3: 11, 12; Exercises 1.4: 18, 19. Due 09/24.
09/15
  1. Read section 1.4 (Outer measures) in Folland. The proofs in this section get quite technical, so start by trying to see the big picture: that it is relatively straightforward to construct outer measures (Proposition 1.10); that outer measures can be used to construct measures (Theorem 1.11); and that if we use a premeasure to construct an outer measure then use that outer measure to construct a measure, the resulting measure will be an extension of the initial premeasure (Theorem 1.14). (Move on, though, to understanding as many details of the proofs as you can.) Send me a question by Wednesday afternoon. Due 09/16.
  2. Exercises 1.3: 7, 9, 11, 12, 13.
09/10
  1. Reread section 1.2 (σ-algebras) of Folland in light of today’s class discussion. (Remember that the case of products of a finite number of spaces is the one that will be of most interest to us this semester, so be sure that the proofs make sense to you in that case at least.)
  2. Read Section 1.3 (Measures) of Folland. Send me an e-mail with one or more questions from the reading by Monday afternoon. (You may also include questions from today’s class if you have any, but you must include something about the reading.) Due 09/14.
  3. Exercises 3 and 4 from the σ-Algebras class exercise handout.
  4. Exercises 1, 2 and 3 from the Gδ Sets class exercise handout.
  5. Folland, Section 1.2, Exercises 3, 4 and 5.
  6. Hand in Exercises 3 and 4 from the σ-Algebras class exercise handout; Folland, Section 1.2, Exercise 5. This assignment will be graded formatively. Due 09/17.
09/08
  1. Read sections 1.1 (Introduction) and 1.2 (σ-algebras) of Folland for discussion on Thursday. (You will find much of Section 1.1 familiar, and you do not need to read it in depth, but it is worth reading Folland’s comments on the properties we would like measurable sets to have.) When you get to the material on product σ-algebras, try to understand the construction at first for a product of 2 spaces, X1 × X2. (This is really the only case we will need.)
  2. Make sure you can answer all questions on the “Limits and Integrals” and “Vitali Sets” class exercise handouts. (On the Notes page.)
09/03
  1. Read the Wikipedia page on Vitali sets. (You will always be able to find links such as this also on the Notes page of this website. Don’t worry about references to “sigma-countability” or “Lebesgue measure;” we will learn about those shortly. You may also skip over the introduction which discusses quotient groups if you are not algebraically inclined: this is an efficient way of seeing that we can define an equivalence relation on real numbers by declaring that x and y are equivalent if x – y is rational. (You should check that this is indeed an equivalence relation.) A Vitali set is a subset of [0,1] which contains exactly one element from each equivalence class.
  2. Review the definition of a (proper) Riemann integral. Explain why the definition requires the interval of integration to be bounded, and the function to be bounded on that interval.
  3. Use the definition of Riemann integral to show that a function which is equal to 0 except at a finite number of points is integrable, with integral 0. Show also that the characteristic function of the set of rational numbers in [0,1] is not Riemann integrable; i.e. fill in the details of Exercise 4 on today’s class handout (on the Notes page.)
  4. In order to verify that we are in communication, send me an e-mail with at least one question you would like answered after today’s class; either something we went over in class that was not clear, or a question arising out of today’s discussion that we did not cover, or something arising out of the above reading and homework. Due 09/07.