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42
SPRING 2012
17.
Measurable Functions, Simple Functions
Proposition 17.1.
If
f,g
:
X
→
R
are
M
measurable, then
f
+
g
and
fg
are
M
measurable.
Proof.
For each
t
∈
R
,
{
x

f
(
x
)+
g
(
x
)
<t
}
=
°
r
∈
Q
±
{
x

f
(
x
)
<r
}
∩
{
x

g
(
x
)
<t
−
r
}
²
.
Since
f
and
g
are
M
measurable, each of the sets on the righthand side is in
M
, so the
lefthand set is as well. Thus
f
+
g
is
M
measurable.
Next, note that
fg
=
(
f
+
g
)
2
−
(
f
−
g
)
2
4
,
so
M
measurability of
fg
follows from the above and Proposition 16.6.
°
Exercise 17.1.
Prove that the
σ
algebra
B
2
of Borel sets in
R
2
is generated by the collection
S
=
{
A
×
R

A
∈
B}
∪
{
R
×
B

B
∈
B}
, where
B
is the
σ
algebra of Borel sets in
R
. (Note
that the coordinate projections
π
i
:
R
2
→
R
de±ned by
π
1
(
x, y
)=
x
and
π
2
(
x, y
)=
y
are
continuous.) Conclude that if
f,g
:
X
→
R
are
M
measurable, then the function
F
:
X
→
R
2
de±ned by
F
(
x
)=(
f
(
x
)
,g
(
x
)) is
M
measurable. Use this to provide an alternative proof of
Proposition 17.1.
Proposition 17.2.
If
f
n
:
X
→
R
is
M
measurable for each
n
∈
N
, then so are
sup
n
f
n
,
inf
n
f
n
,
lim sup
n
f
n
,
lim inf
n
f
n
.
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 Spring '09

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