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Mathematics 105, Spring 2004 — Problem Set IV
For the remainder of the semester we will follow the text
A Concise Introduction
to the Theory of Integration
(second edition) by Stroock. Problem, chapter, section,
and page numbers refer to that text unless otherwise indicated.
For Friday February 27
: You should have already read
§
2.0 and begun reading
§
2.1. Finish studying
§
2.1. Also read Lemma 1.1.1 and its proof; the whole theory
of Lebesgue integration relies on this (semitrivial) lemma.
Solve problems 2.1.18,19 of Stroock, and the following problems:
IV.A
Let
{
I
n
}
be any
ﬁnite
set of open intervals that covers [0
,
1]
∩
Q
. Show that
∑
n

I
n
 ≥
1. Explain why this does
not
prove that

[0
,
1]
∩
Q

e
≥
1.
IV.B
Show that for any Lebesgue measurable sets
A, B
⊂
R
n
,

A
∪
B

+

A
∩
B

=

A

+

B

.
IV.C
Let
A
⊂
R
n
and suppose that

A

e
= 0. Prove that for any set
B
⊂
R
n
,

A
∪
B

e
=

B

e
.
IV.D
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This note was uploaded on 05/07/2008 for the course MATH 105 taught by Professor Michaelchrist during the Spring '04 term at Berkeley.
 Spring '04
 MichaelChrist
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