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Math 320 Measure Theory and Integration
Assignment 3: Measurable Functions and Their Unexpected Properties
The due date for this assignment is 9/30
1. Consider the Cantor function
: [0
;
1]
!
[0
;
1]
x
2
C
, the Cantor set, we can write
x
=
P
1
i
=1
x
n
3
n
with
x
0
= 0
or
2
and set
(
x
) =
1
X
i
=1
x
n
3
n
!
:=
1
X
i
=1
x
n
2
1
2
n
.
Extend
(
x
)
to
[0
;
1]
by
(
x
) = sup
f
(
y
) :
y
2
C; y < x
g
for each
x
2
[0
;
1]
n
C
.
(A) Show that
is increasing and continuous but
0
= 0
almost everywhere on
[0
;
1]
.
(B) Show that
maps
C
onto
[0
;
1]
.
2. (A) Use problem 1 (or otherwise) show that there exists a Lebesgue measurable function
f
and a Lebesgue measurable set
E
such that
f
(
E
)
is not Lebesgue measurable. (Hint:
Consider
f
(
x
) =
x
+
(
x
)
.) (B) Show that there exists a Lebesgue measurable function
f
and a Lebesgue measurable set
E
such that
f
1
(
E
)
is not Lebesgue measurable. (C)
Show that there exists a Lebesgue measurable set that is not a Borel set.
3. (A) Show that if
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 Fall '10
 GREGORYMARGULIS
 Math

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