Math 426: Homework 1 Solutions
Mary Radclie
due 9 April 2014
In Bartle:
2B. Show that the Borel algebra B is also generated by the collection of all
half-open intervals (a, b] = cfw_x R | a < x b. Also show that B is
generated by the collection of all hal
Math 426: Homework 3
Mary Radclie
due 23 April 2014
In Bartle: 5C, 5E, 5I, 5P, 5Q, 5R, 5T, 4O
Note: Problems 5A-5H are all very straightforward but USEFUL properties
of the integral. You should read them, verify that you know how to prove them,
and assimi
Name:
Math 426/576: Midterm Solutions
2 May 2014
Turn o and put away your cell phone.
No notes or books are permitted during this exam.
No calculators or any other devices are permitted during this exam.
Read each question carefully, answer each question
Math 426: Homework 7
Mary Radclie
due 30 May 2014
In Bartle: 10G, 10H, 10K, 10L, 10P, 10R
Read through 10A, B, C, E, some basic properties of Cartesian products.
1. Given (X, F) and (Y, G) measurable spaces. Let Z0 be the set of nite
unions of sets of the
Math 426: Homework 5 Solutions
Mary Radclie
due 14 May 2014
In Bartle:
1
6H. Let X = Z+ , and let be the measure on X which has measure n2 at
the
point n. Show that (X) < . Let f be dened on X by f (n) = n.
Show that f Lp if and only if 1 p < 2.
Solution.
Math 426: Homework 6
Mary Radclie
due 21 May 2014
In Bartle:
7G. If a sequence cfw_fn converges in measure to a function f , then every subsequence of cfw_fn converges in measure to f . More generally, if cfw_fn is
Cauchy in measure, then every subsequ
Math 426: Homework 6
Mary Radclie
due 21 May 2014
In Bartle: 7G, 7I, 7Q, 7R, 7V, 7W
Read through 7A-7F, 7J-7M, some examples of functions that converge in
various dierent ways, and make sure you can prove them.
Read through 7N, 7O, some variants on our fa
Math 426: Homework 5
Mary Radclie
due 14 May 2014
In Bartle: 6H, 6I, 6N, 6P, 6Q, 6R, 6T
Not to turn in, but denitely to know: 6C, 6D, 6E, 6F, 6K, 6L, 6U
Also, complete the following:
1. WARNING: In the following exercise, L has nothing to do with integrat
Math 426: Homework 3
Mary Radclie
due 23 April 2014
In Bartle:
m
4A. If the simple function M + has the representation = k=1 bk Fk ,
m
where bk R and Fk F, prove that d = k=1 bk (Fk ).
Proof. For 1 k m, let k = bk Fk + 0X\Fk . Then k is in standard
m
form
Math 426: Homework 2 Solutions
Mary Radclie
due 16 April 2014
In Bartle:
9B. Show that the family G of all nite unions of sets of the form (a, b), (, b),
(a, ), (, ) is not an algebra of sets in R.
Solution. Consider E = (, 0) (1, ) G. Note that R\E = [0,
Math 426: Homework 4
Mary Radclie
due 2 May 2014
In Bartle:
5C. If f L(X, F, ) and g is a F-measurable real-valued function such that
f (x) = g(x) almost everywhere, then g L(X, F, ) and f d = g d.
Proof. Let h(x) = |f (x) g(x)|. Then h M + (X, F), and h
Math 426: Homework 1
Mary Radclie
due 9 April 2014
In Bartle: 2B, 2K, 3E, 3F, 3H, 3T, 3U, 3V
Please consider (but do not turn in): 2D, 2E, 3I, 3J. These four problems all
deal with lim inf and lim sup of a collection of subsets of X. You will be responsib
Math 426: Homework 3
Mary Radclie
due 23 April 2014
In Bartle: 4A, 4B, 4G, 4K, 4L, 4M, 4S, 4T
Also, complete the following:
1. (a) Show, by example, that if cfw_fn M + is a sequence of functions with
fn (x) fn+1 (x) (i.e., a decreasing sequence) with fn
Math 426: Homework 2
Mary Radclie
due 16 April 2014
In Bartle: 9B, 9E, 9S
Notation: Throughout, refers to Lebesgue measure, and we assume the
Axiom of Choice, forever and ever.
1. Prove Theorem 3 in the additional notes.
2. Complete the proof of Theorem 4