Math 8100 Assignment 4
Lebesgue Integration
Due date: Thursday the 10th of October 2013
1. A sequence cfw_fk of integrable functions on Rn is said to converge in measure to f if for every > 0,
lim m(cfw_x Rn : |fk (x) f (x)| ) = 0.
k
(a) Prove that if fk
Math 8100 Exam 1
Thursday 26th of September 2013
Answer any FIVE of the following six problems
1. Let E Rn . Recall that the denition of the Lebesgue outer measure of E , m (E ) is given by
|Qj |
m (E ) = inf
j =1
where the inmum is taken over all countab
Math 8100 Assignment 1
Lebesgue measure and outer measure
Due date: Tuesday the 27th of August 2013
1. The Cantor set C is the set of all x [0, 1] that have a ternary expansion x = k=1 ak 3k with
ak = 1 for all k . Thus C is obtained from [0, 1] by removi
Math 8100 Final Exam
Thursday 12th of December 2013
1. Let (X, M, ) be a measure space and g be a non-negative measurable function on X . Show that
(E ) =
g d
E
denes a measure on X and that for any non-negative measurable function f we have
f d =
f g d.
Math 8100 Exam 2
Thursday 14th of November 2013
1. (a) Which of the following statements (i)-(iv) are true, and which are false? Justify all the negative
answers by a counterexample and all positive answers with a proof.
(i) L2 (R) L3 (R)
(ii) L3 (R) L2 (
Math 8100 Assignment 2
Lebesgue measurable sets and functions
Due date: Tuesday the 10th of September 2013
1. Let E be the subset of [0, 1] which consists of all numbers which do not have the digit 4 appearing in
their decimal expansion. Find m(E ).
2. Re
Math 8100 Assignment 3
Lebesgue Integration
Due date: Thursday the 19th of September 2013
1. Let E be a Lebesgue measurable subset of Rn with m(E ) < and > 0. Show that there exists a
set A that is a nite union of closed cubes such that m(E A) < .
[Recall
Math 8100 Assignment 7
Due date: Thursday 5th of December 2013
1. (a) Let X be an uncountable set and A = cfw_E X : E is countable or E c is countable.
i. Verify that A is a -algebra, called the -algebra of countable or co-countable sets.
ii. Verify that
Math 8100 Assignment 5
Due date: Tuesday 22nd of October 2013
1. Prove the following properties of L = L (Rn ).
(a) If f and g are measurable functions on Rn , then f g
f
1
g
1
.
(b)
(c) L
is a Banach space.
is a norm on L .
(d) fn f
0 i there exists E R
Mathematics Department
The University of Georgia
Math 8100 Midterm
October 24, 2012
1. If the set E R has measure zero, then the set cfw_x2 : x E has measure zero. Prove
the statement.
Hint: Let S = cfw_x2 , x E . Show that Sn = S (0, n2 ) has measure ze