Math 676
Problem Set #4
Measurable Functions
Solutions
1. Suppose that A
E
B where A and B are measurable sets of
nite measure with m(A) = m(B ). Let " > 0 be given. There is an
open set O
B with m(O B ) " since B is measurable. The set
O A is measurable
Math 676
Problem Set #7
The Lebesgue Integral, Part II
Due March 28, 2012
1. (page 93, problem 18) You know what to do!
2. (page 94, problem 21)
(a) The function g (y ) is measurable viewed as a function on Rd Rd
by Corollary 3.7. The function f (x y ) is
Math 676
Problem Set #8
Solutions
1. (page 93, problem 19) Observing that
Z
m(E ) =
E
(x) dx
Rd
we have
Z
1
m(E )d =
0
Z
1
0
=
Z
Rd
=
Z
Rd
=
Z
Rd
Z
Z
(x) dx
E
0
Z
jf (x)j
d
(x) d
!
E
Rd
1
dx
1d
0
dx
jf (x)j dx
where in the second and third lines we used T
Math 676
Problem Set #9
Integration and Dierentiation, Part II
Solutions
R
1. (a) Suppose that ' is an integrable function with '(x) dx = 1, and
let K (x) = d '(x= ). We compute
Z
Z
d
K (x) dx =
'(x= ) dx
and use the change of variables y = x= and the sca
Math 676
Problem Set #6
The Lebesgue Integral, Part I
Due February 29, 2012
1. (page 91, problem 6)
(a) Let g (x) be a bump function with 0 g (x) 1, g (x) = 1 on
jxj
1 and g (x) = 0 for jxj
2. Let gn (x) = ng (n3 (x n).
Then
Z
Z
gn (x)dx = n g (n3 (x n) d
Math 676
Problem Set #5
Measure and the Integral
Solutions
1. This problem concerns the inequality relating geometric and arithmetic
means for nonnegative numbers x1 ;
; xd :
x1 +
xd )1=d
(x1
+ xd
(1)
d
First, we show that the inequality holds when d = 2k
MA 676
Chad Linkous
Homework # 1
January 23, 2012
1. Prove that the Cantor set C constructed in the text is totally disconnected and perfect. In other
words, given two distinct points x, y C , there is a point z C that lies between x and y , and yet
/
C h
Math 676
Problem Set #2
Jordan Measure
Solutions
1. This problem concenrs properties of elementary measure. In what follows, E and F denote elementary subsets of Rd .
(a) (Monotonicity) Suppose E and F are elementary sets and E F .
The sets E \F and F E a
Math 676
Problem Set #3
Measure, Continued
Solutions
1. Suppose that E is a subset of Rd and On = fx : d(x; E ) < 1=ng.
(a) Suppose that E is compact (hence measurable). Note that On+1
On . We claim that \1 On = E . Clearly E \1 On so suppose
n=1
n=1
that
Math 676
Problem Set #10
Integration and Dierentiation, Part III
Solutions
1. (Stein and Shakarchi, p. 147, #11) For a; b > 0, let
xa sin(x b ); 0 < x 1
:
0
x=0
f (x) =
Note that
f 0 (x) = axa
1
sin(x b )
bxa
b1
cos(x b )
axa
+ bxa
b1
:
obeys the estimate