### MATH 563 - HOMEWORK PROBLEMS - Fall 2014

Problems from Stein-Shakarki's book.

Assignment 4: (you can work in teams like with the first assignment, only difference is that this time all teams are doing the same assignment)

1. Exercise 24 in Chapter 3 (Lebesgue decomposition of an increasing and bounded function).
2. Exercise 11 in Chapter 6 (Decomposition of the measure associated to an increasing and bouded function into an absolutely continuous part, a jump part, and a singular part).

Assignment 3: (you can work in teams like with the first assignment, only difference is that this time all teams are doing the same assignment)

1. Show H\"older's inequality and the triangle inequality in L^p.
2. Show that L^p is a complete normed space.

Assignment 2: (you can work in teams like with the first assignment, only difference is that this time all teams are doing the same assignment)

1. Show that SS and Tao's definition of Lebesgue measurable sets coincides with Caratheodory's definition (Chapter 6, Exercise 3 in SS p.312).
2. Show that the image under a linear transformation on R^d of a Lebesgue measurable set on R^d is a Lebesgue measurable set. (Chapter 1, Exercise 8 in SS p.39).
3. Show the Borel-Cantelli Lemma. (Chapter 1, Exercise 16 in SS p.42).

Group Work 1:
Each team will submit a report. We assigned the 4 lemmas (or a variation) stated in my first electronic message (I remember for sure what the assignment was for Kenney and Audrey, maybe for the other three teams is a permutation of what I am writing. We talked about Cantor sets, and I would like each team to do one Cantor set exercises in Stein-Shakarchi (SS) pages 37-39. (The basic topological definitions that we need are listed under "Open, closed and compact sets" in pages 2-3 Stein-Shakarchi.)

1. Kenney and Audrey: open sets is R can be written in a unique way as a disjoint union of at most countable open intervals. Note that if we replace R by R^d and open intervals by open boxes the statement is false. The correct statement replaces open intervals by open connected sets. Do also Exercise 4. Cantor-like sets (and take a pick at Exercise 9).
2. Michael, Jacob and Patrick: if a subset E of R is the union of finitely many disjoint intervals {I_j: 1<= j <+N} and is also the union of finitely many disjoint intervals {J_k: 1<= k <= M} then the sum of |I_j| is equal to the sum |J_k|. Do also Exercise 2.
3. Mostafa and Wenjing: if E in R is a finite union of intervals then E is a finite union of disjoint intervals. Similarly if E in R^d is a finite union of boxes then E is a finite union of disjoint boxes. Do also Exercise 3.
4. Chris and John: If B is a box that is the union of finitely many disjoint boxes {B_j: 1<= j <=N} then the volume of B is the sum of the volumes the B_j. Do also Exercise 1.