Math 514
Solutions to HW #1
February 5, 2008
Exercises 1, 5, 6, chapter 1, section 1, p. 7 (in Cohn)
1. Let A denote the
algebra on R generated by the singletons. That
means, A is a
algebra, A contains all the singletons, and for any other
algebra A0 on R
Math 514
Solution to HW #5
Spring, 2008
Problem: Show that for f : (X; A) ! R; the property that
(8t 2 R) f
1
(ftg) 2 A
does not imply that f is measurable.
Solution: We give an example on the real line with respect to Lebesgue
ll
measure on [0; 1]. That
Math 514
Solutions to HW #10
May, 2008
section 3.2 # 14, section 3.3, # 7, section 4.2, # 12, section 4.4,
#5
section 3.2 # 14
Suppose that k k is a seminorm that gives convergence in measure on the
measure space f[0; 1] ; g ; as described in the problem:
Math 514
Solutions to HW #9
May 6, 2008
Section 3.1, problems 2, 4, 5, 6
4. We suppose that
1
XZ
jfn f j d < 1:
n=1
X
We want to show that fn ! f a:e: For notational convenience, let write
s
gn = fn f: So we are given
1
XZ
jgn j d < 1;
X
n=1
and we want t
Math 514
Solution to HW #4
Spring, 2008
Problem: Show that the range of Lebesgue measure on a set A R is
an interval.
Solution: Fix a set A R of positive Lebesgue measure. We want to show
that
f (B) j B A; B is Lebesgue measurableg
is an interval.
Conside