Serge Ballif
MATH 501 Homework 2
October 3, 2007
In problems 13, (
X,
M
,μ
) denotes a measure space.
1.
Let
(
E
k
)
k
∈
N
⊂ M
. Deﬁne
lim inf
E
j
=
∞
[
i
=1
∞
\
k
=
i
E
k
and
lim sup
E
j
=
∞
\
i
=1
∞
[
k
=
i
E
k
Prove the following.
(a)
μ
(lim inf
E
j
)
≤
lim inf
j
→∞
μ
(
E
j
)
.
(b) If
μ
(
S
∞
j
=1
E
j
)
<
∞
, then
lim sup
j
→∞
μ
(
E
j
)
≤
μ
(lim sup
E
j
)
. Show the assumption
μ
(
S
∞
j
=1
E
j
)
<
∞
is necessary.
(c)
BorelCantelli lemma.
If
∑
∞
j
=1
μ
(
E
j
)
<
∞
, then
μ
(lim sup
E
j
) = 0
.
(a) Recall that lim inf
j
→∞
x
j
= lim
j
→∞
±
inf
k
≥
j
x
k
²
.
Claim 1.
μ
(
S
∞
i
=1
T
∞
k
=
i
E
k
) = lim
i
→∞
μ
(
T
∞
k
=
i
E
k
).
Proof.
Let
F
i
=
T
∞
k
=
i
E
k
. We have the increasing chain of subsets
F
i
⊂
F
i
+1
for
all
i
. Hence, by continuity from below
μ
(
∪
∞
j
=1
F
j
) = lim
j
→∞
μ
(
F
j
).
For each
k
we know by monotonicity that
μ
(
T
∞
k
=
j
E
k
)
≤
μ
(
E
k
).
Hence
μ
(
T
∞
k
=
j
E
k
) is a lower bound for
μ
(
E
k
). Therefore,
μ
(
T
∞
k
=
j
E
k
)
≤
inf
k
≥
j
μ
(
E
k
). Now
we take the limit as
j
→ ∞
to get
lim
j
→∞
μ
∞
\
k
=
j
E
k
!
=
μ
(lim inf
E
j
)
≤
lim inf
j
→∞
μ
(
E
j
)
.
(b) Recall that lim sup
j
→∞
x
j
= lim
j
→∞
±
sup
k
≥
j
x
k
²
.
Claim 2.
μ
³
T
∞
j
=1
S
∞
k
=
j
E
k
´
=
lim
j
→∞
μ
³
S
∞
k
=
j
E
k
´
.
Proof.
Let
G
j
=
S
∞
k
=
j
E
k
. We have a decreasing chain of subsets
G
j
⊃
G
j
+1
for
all
j
. Also, we know that
μ
(
G
j
)
<
∞
. Hence, by continuity from above we know
that
μ
(
T
∞
j
=1
G
j
) = lim
j
→∞
μ
(
G
j
).
For each
k
we know by monotonicity that
μ
(
E
k
)
≤
μ
(
S
∞
k
=
j
E
k
). Hence,
sup
k>j
μ
(
E
k
)
≤
μ
(
S
∞
k
=
j
E
k
). Now we take the limit as
j
→ ∞
to get
lim
j
→∞
μ
∞
[
k
=
j
E
k
!
=
μ
∞
\
j
=1
∞
[
k
=
j
E
k
!
≤
lim
j
→∞
±
sup
k
≥
j
μ
(
E
k
)
²
.
To see that the assumption
μ
(
S
∞
j
=1
E
j
)
<
∞
is necessary, we look at the in
tervals
E
j
= (
j,
∞
) where our measure is the Lebesgue measure on
R
. Then
lim sup
E
j
=
T
∞
i
=1
S
∞
k
=
i
E
k
=
∅
.
Then lim sup
j
→∞
μ
(
E
j
) =
∞
>
0 =
μ
(
∅
) =
μ
(lim sup
E
j
)
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MATH 501 Homework 2
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 Fall '08
 WYSOCKI,KRZYSZTOF
 Math, measure, Lebesgue measure, Lebesgue integration, lim sup Ej, Hj Ej

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