Introductory Analysis 2–Spring 2010
Homework 9
Due April 7, 2010
Wrapping up Chapter 3
1. Recall the exercise from Exam 2.
Let
˜
‘
be a function from open subintervals of
R
to [0
,
∞
] so that if
I
= (
a, b
),
∞ ≤
a
≤
b
, then
˜
‘
(
I
)
∈
[0
,
∞
). Assume it sends the empty interval to 0. If
A
⊂
R
define
μ
*
(
A
) = inf
{
X
n
∈
N
˜
‘
(
I
n
) :
I
1
, I
2
, . . .
open intervals,
A
⊂
∞
[
n
=1
˜
‘
(
I
n
)
}
.
In the exam you were asked to prove this is an outer measure. I assume that done.
(a) Give an example to show that
μ
*
(
I
) =
˜
‘
(
I
) for open intervals
I
may fail to hold.
(b) Let
f
:
R
→
R
be increasing (
x < y
⇒
f
(
x
)
≤
f
(
y
)) and continuous.
If
I
= (
a, b
) is an open interval, define
˜
‘
(
I
) =
f
(
b
)

f
(
a
).
Prove:
Defining
μ
*
as above, all Borel sets are measurable.
Is it necessary for
f
to be
continuous? (Interpret
f
(
∞
) = lim
x
→∞
f
(
x
),
f
(
∞
) = lim
x
→∞
f
(
x
).)
(c) Suppose that instead of open intervals we use half open intervals of the form (
a, b
] with
∞
< a
≤
b <
∞
, assume
we have
f
:
R
→
R
increasing and define
˜
‘
((
a, b
]) =
f
(
b
)

f
(
a
). We define
μ
*
(
A
) as above, but using only these
bounded halfopen intervals; that is if
A
⊂
R
we define
μ
*
(
A
) to be the infimum of the set of all sums of the form
∑
∞
n
=1
˜
‘
(
I
n
), where
{
I
n
}
ranges over all sequences of halfopen intervals of the form (
a, b
] that cover
A
. The proof
that this is an outer measure is the same as before; you might want to do it for your own benefit. Prove: If
f
is
continuous from the right; that is,
f
(
a
) = lim
x
→
a
+
f
(
x
) for all
x
∈
R
, then all Borel sets are measurable. Is this
condition necessary?
Solution.
I sometimes do assign exercises that involve some investigative work, where I add questions whose answers
I think I know, but haven’t quite worked out. This is one of them. I was absolutely sure that the question about the
continuity of
f
in parts b, c would be “yes; it is necessary.” It turns out it isn’t.
Lets start answering the questions.
The fact that any such
μ
*
is an outer measure is quite straightforward and was discussed in class. So we can turn to
the questions of the homework.
(a) Almost any example at random will do. For example suppose one defines
˜
‘
(
I
) = 1 if
I
is an unbounded interval,
˜
‘
(
I
) = 0 otherwise. Then, for example, if
I
=
R
we have
˜
‘
(
I
) = 1. But
R
⊂
S
∞
n
=1
(

n, n
) and since
˜
‘
((

n, n
)) = 0 we
get
∑
∞
n
=1
˜
‘
((

n, n
)) = 0, implying
μ
*
(
I
) = 0.
(b) A lot of what I do here is also applicable to part (c), and will not be repeated. In the first place we should remember
a few facts about increasing functions, such as
If
f
:
R
→
R
is increasing then for every
x
∈
R
both
f
(
x

) = lim
y
→
x

f
(
y
),
f
(
x
+) = lim
y
→
x
+
f
(
y
) exist and we have
if
a < x < b
,
f
(
a
)
≤
f
(
x

)
≤
f
(
x
)
≤
f
(
x
+)
≤
f
(
b
)
.
In addition,
f
(
∞
)
, f
(
∞
) can be defined (if one wants to, and here we want) by
f
(
∞
) = lim
x
→∞
f
(
x
),
f
(
∞
) =
lim
x
→∞
f
(
x
). We could have
f
(
∞
) =
∞
or
f
(
∞
) =
∞
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 Speinklo
 CN, Lebesgue measure, open intervals

Click to edit the document details