Mathematics 105 — Spring 2004 — M. Christ
Problem Set 9 — Solutions to Selecta
IX.A
Consider the measure space (
R
1
,
B
R
1
, λ
) where
λ
denotes Lebesgue measure.
Consider the measurable functions
f
n
(
x
) =
1
n
χ
[0
,n
]
. Show that
f
n
→
0 uniformly on
R
. Show that
f
n
dλ
→
1. Explain why this does not contradict any of our three
basic convergence theorems.
Solution.
That
f
n
→
0 uniformly is obvious: Given
ε >
0, choose an integer
N >
1
/ε
.
Then whenever
n
≥
N
, for any
x
∈
R
, either

f
n
(
x
)

0

=
n

1
< ε
, or

f
n
(
x
)

0

=

0

0

< ε
. Since
N
does not depend on
x
, this establishes uniform
convergence.
It’s equally obvious that
f
n
dλ
=
n

1
λ
([0
, n
]) =
n/n
= 1.
On the
other hand,
lim
n
→∞
f
n
dλ
=
0
dλ
= 0, which is not equal to 1 = lim
n
→∞
f
n
dλ
.
This doesn’t contradict the monotone convergence theorem, because the hypoth
esis that
f
n
≤
f
n
+1
for all
n
is violated.
There’s no violation of the dominated convergence theorem. If
f
n
(
x
)
≤
g
(
x
) for
all
x, n
, then of course
g
(
x
)
≥
sup
n
f
n
(
x
), and if
k

1
< x < k
for some positive
integer
k
then sup
n
f
n
(
x
) =
k

1
. Thus
g
(
x
)
≥
k

1
for all
x
∈
(
k

1
, k
). Therefore
for any “Lebesgue dominator”
g
,
g dλ
≥
∑
∞
k
=1
k

1
λ
((
k

1
, k
)) =
∑
∞
k
=1
k

1
= +
∞
.
Therefore there is no
integrable
Lebesgue dominator, so the Dominated Convergence
Theorem doesn’t apply.
The hypotheses of Fatou’s lemma
are
satisfied — but there’s no contradiction since
Fatou’s lemma doesn’t assert equality, only the inequality
0
dλ
≤
lim
n
→∞
1 = 1.
IX.B
Let (
E,
A
, μ
) be any measure space. Suppose that
f
:
E
→
R
1
is a measurable
function (which never takes on the values
±∞
). Recall that for any
Borel
measurable
set
S
⊂
R
1
, the set
f

1
(
S
) belongs to
A
.
(You need not prove this.).
Define
ν
(
S
) =
μ
(
f

1
(
S
)) for all
S
∈ B
R
1
Show that
ν
is a measure on the Borel
σ
algebra
B
R
.
More generally, suppose that
f
:
E
→
R
n
is a measurable mapping, in the sense
that
f

1
(
S
)
∈ A
for all
S
∈ B
R
n
. Show that the above recipe defines a measure on
B
R
n
.
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 Spring '04
 MichaelChrist
 Math, FN, Dominated convergence theorem, Lebesgue measure, Lebesgue integration

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