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Math 8601: REAL ANALYSIS.
Fall 2010
Homework #6
(due on Wednesday, November 24).
50 points are divided between 5 problems, 10 points each.
#1
(BorelCantelli Lemma). Let (
X,
M
,μ
) be a measure space, and
{
A
n
}
be a sequence of
sets in
M
. Show that
if
∞
X
n
=1
μ
(
A
n
)
<
∞
,
then
μ
±
lim sup
n
→∞
A
n
²
= 0
,
where
lim sup
n
→∞
A
n
:=
∞
³
k
=1
∞
[
n
=
k
A
n
.
#2.
The
symmetric diﬀerence
of two sets
A
and
B
is deﬁned as
A
Δ
B
= (
A

B
)
∪
(
B

A
) = (
A
∪
B
)

(
A
∩
B
)
.
Let
A
1
,A
2
,
···
be a sequence of measurable sets in a measure space (
X,
M
,μ
) satisfying
lim
n
→∞
sup
j,k
≥
n
μ
(
A
j
Δ
A
k
) = 0
.
Show that there exists a measurable set
A
, such that
lim
k
→∞
μ
(
A
k
Δ
A
) = 0
.
#3.
Let
f
(
x
) be a right continuous function on
R
1
, i.e.
lim
x
→
x
0
,x>x
0
f
(
x
) =
f
(
x
0
)
for all
x
0
∈
R
1
.
(a).
Show that
f
is measurable.
(b).
Show that
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 Fall '08
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