Jerey Hellrung
Wednesday, October 12, 2005
Math 245A, Homework 01
Chapter 1, # 3, 4, 5, 7, 8, 12, 14, 15
3. Let M be an innite -algebra.
a. M contains an innite sequence of disjoint sets.
b. card(M) c.
Solution
a. Construct the sequences
cfw_En
n=0
and
c
Jerey Hellrung
Wednesday, November 16, 2005
Math 245A, Homework 06
Chapter 2, # 46, 49, 50, 54, 56, 60, 61, 64
46. Let X = Y = [0, 1], M = N = B[0,1] , = Lebesgue measure, and = counting measure. If D =
cfw_(x, x) : x [0, 1] is the diagonal in X Y , then
Jerey Hellrung
Wednesday, November 09, 2005
Math 245A, Homework 04
Chapter 2, # 9, 10, 12, 13, 14, 15, 16, 19, 20
9. Let f : [0, 1] [0, 1] be the Cantor function (1.5), and let g (x) = f (x) + x.
a. g is a bijection from [0, 1] to [0, 2], and h = g 1 is c
Jerey Hellrung
Wednesday, October 19, 2005
Math 245A, Homework 02
Chapter 1, # 13, 17, 19, 25, 26, 27, 29
13. Every -nite measure is seminite.
Solution
Let be a -nite measure on (X, M); then we can decompose X = n=1 En for En M such that
(En ) < . Without
Jerey Hellrung
Wednesday, October 26, 2005
Math 245A, Homework 03
Chapter 1, # 18, 30, 31, 33
Chapter 2, # 2, 3, 4, 8
18. Let A P (X ) be an algebra, A the collection of countable unions of sets in A, and A the collection
of countable intersections of set
Jerey Hellrung
Wednesday, November 16, 2005
Math 245A, Homework 05
Chapter 2, # 21, 23, 25, 26, 27, 34, 39, 40, 42, 44
21. Suppose fn , f L1 and fn f a.e. Then
Solution
Suppose
|fn f | 0 i
|f |. (Use Exercise 20.)
|fn f | 0. Then
|fn | |fn f | + |f |
so