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. Als
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 t
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 ex
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
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
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 general
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 p
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 th
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
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.
So
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.
Pr
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
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
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 t