| Week: | Aug 28 |
| Material: |
Chapter I sections 2.1, 2.2 |
| Homework: | All exercises from sections 2.1 and 2.2 and read carefully. Homework to be graded: Define the Lebesgue function geometrically, as in class. Show 1) f extends to a continuous function on [0,1] 2) Show that f is not differentiable at x=0. |
| Remarks: | In
general: The homework exercises will always be ALL the exercises from
the sections discussed that week. Note: there is an answer section at
the end of the book. |
| Week: | Sep 4 |
| Material: |
section 2.3, 2.4 Example of a non-measurable set. |
| Homework: | exercises from section 2.3 . Homework to be graded: Let f(x)=x+ t mod 1, where t is irrational. The set of labels of the orbits of f is denoted by Z. Show 1) m^*(f^k(Z))=m^*(Z). Hint: proposition 2.8 2) Z is not measurable |
| Remarks: |
| Week: | Sep 11 |
| Material: | 2.5, 3.1, 3.2, 3.3, 3.4 Example of a measurable set which is not Borel. |
| Homework: | All exercises form these sections Homework to be graded: 1) Construct an injective non measurable function f:[0,1] to R. 2) If f,g: R to R are both measurable, does this imply that also f composed with g is measurable? |
| Remarks: |
| Week: | Sep 18 |
| Material: | 4.1, 4.2, 4.3 up to Theorem 4.19 |
| Homework: | |
| Remarks: | Midterm I will be held on Oct 19 in class. The material will be announced later. There will be a review on Oct 17. No Homework this week |
| Week: | Sep 25 |
| Material: | 4.3, 4.4, 4.5 |
| Homework: | Home work to be graded: 1) Let f_n be a nonnegative decreasing sequence of integrable functions with S f_n converges to 0 (The S stands for integral). Show f_n converges to 0 almost everywhere. 2) Let f, g be integrable and f bounded. Show fg is integrable. Give an example of a pair of piecewise constant functions, both integrable functions but fg is not integrable. |
| Remarks: | Midterm I will be held on Oct 19 in class. The material will be announced later. There will be a review on October 17. |
| Week: | Oct 2 |
| Material: | 4.6, 5.1 |
| Homework/Project: | Homework to be graded: Let f be a function defined on [0,1]: on [0,1/5) f(x)=5/2 x on [1/5, 1/2) f(x)=5/3 (x -1/2)+1 on [1/2, 1] f(x)=2x mod 1 Let h be a step function. Define a new step function by Th(x)= sum_{y\in f^{-1}(x)} 1/Df(y) h(y) where Df is the derivative of f. 1) Show there is a step function h_0\>0 with Th_0=h_0 and \int h_0=1 where \int means integral. 2) Show \int Th =\int h for every step function The statement means that the total amount of sand does not change. 3) Show that for every step function h with \int h=1: T^n h converges to h_0 in L^1, i.e. \int |T^n h-h_0| converges to 0. 4) Show hat for every integrable f with \int f=1: T^n f converges to h_0 in L^1, i.e. \int |T^n f-h_0| converges to 0 The due date for this homework is Thursday November 3. |
| Remarks: | Midterm I will be held on Oct 19 in class. The material for the midterm is everything from the schedule up to and included 4.6. There will be a review on October 12 |
| Week: | Oct 9 |
| Material: | no new material |
| Homework: | |
| Remarks: | Midterm I will be held on Oct 19 in class. The material for the midterm is everything from the schedule up to and included 4.6. There will be a review on October 12. Prepare questions. |
| Week: | Oct 16 |
| Material: | 5.1, 5.2 |
| Homework | Homework to be graded: Do the usual exercises from the book. No homework collection for this week. |
| Remarks: | Midterm I will be held on Oct 19 in class. The material for the midterm is everything from the schedule up to and included 4.6. |
| Week: | Oct 23 |
| Material: | 5.2, 5.3 |
| Homework: | 1) Construct a sequence of functions in L^2([0,1]) which is Cauchy in L^1 but not Cauchy in L^2. 2) Let K be closed subspace of L^2([0,1]). Show that the orthogonal projection onto K is a continuous linear map. 3) Show that L^2([0,1]) is dense in L^1([0,1]), i.e. for every f in L^1([0,1]) there exists a sequence of functions f_n in L^2([0,1]) such that || f_n-f||_1 to 0. 4) Show that there does not exists an inner product on L^1([0,1]) which induces the L^1-norm. |
| Remarks: |
|
| Week: | Oct 30 |
| Material: | 6.1, 6.2, 6.3 |
| Homework: | 1) Show that the closed unit disk is in the sigma-field of the plane. 2) Show that the open unit disk is in the sigma-field of the plane. |
| Remarks: | The average for the first midterm is 14/40. The average corresponds more or less to a B. If you got above the average you did well. If below the average you have to be more careful. |
| Week: | Nov 6 |
| Material: | 6.4, 7.2 |
| Homework: | Give a proof of Corollary 6.11. |
| Remarks: |
The second midterm will be held in class on Tuesday November 28 The material will be up to and included section 7.2. There will be a review on Thursday November 23. |
| Week: | Nov 13 |
| Material: | 7.2, 7.3 |
| Homework: | Let p >1/2. and mu_p the Bernouili measure 1) Let x=k/2^n a dyadic rational. Describe mu_p([x, x+delta x] when delta x to 0. 2) Let x=k/2^n a dyadic rational. Describe mu_p([x-delta x,x] when delta x to 0. 3) Let x in E_p. Describe mu_p([I_n] when |I_n| to 0 and I_n a dyadic interval. 4) The distribution function of mu_p is Holder. What is the optimal Holder-exponent? |
| Remarks: | The second midterm will be held in class on Tuesday November 28 The material will be up to and included section 7.2. There will be a review on Thursday November 23. |
| Week: | Nov 20 |
| Material: | 7.3 |
| Homework: | For each p construct an iterated function system whose attractor is the the graph of the distribution function of the Bernoulli measure mu_p. Include a proof. This homework is due in the week of November 28. |
| Remarks: | The second midterm will be held in class on Tuesday November 28 The material will be up to and included section 7.2. There will be a review on Thursday November 23. The threshold for The Big Homework is 30/40. If you got at least 30 you do not need to do the final exam. |
| Week: | Nov 27 |
| Material: | |
| Homework: | |
| Remarks: | The second midterm will be held in class on Tuesday November 28. |
| Week: | Dec 4 |
| Material: | 7.3.4 |
| Homework: | |
| Remarks: | Final exam: Material: everything from the schedule up to and included 7.3.4. Date: Dec 21 Time: 11:15-13:45 Room: Lgt Eng 154 Thursday: Review. Average Midterm II: 26 The average corresponds more or less with a B. |