Measure and Integration
Problems
1
Problem 1.1
Let
C
be a collection of subsets of the set
Ω
. For each
A
∈
σ
(
C
)
show that there is a countable subcollection
C
A
of
C
such that
A
∈
σ
(
C
A
)
.
Problem 1.2
Let
(Ω
,
F
)
and
(Ω
0
,
F
0
)
be measurable spaces and
f
: Ω
→
Ω
0
a function.
(a) Prove that
f

1
(
F
0
)
is a
σ
algebra on
Ω
.
(b) Show that
f
(
F
)
need not be a
σ
algebra even if
f
is surjective.
2
Problem 2.1
Find a measurable space
(Ω
,
F
)
carrying ﬁnite measures
μ
and
ν
such that
μ
(Ω) =
ν
(Ω)
but such that
{
A
∈ F
:
μ
(
A
) =
ν
(
A
)
}
is not a
σ
algebra.
Problem 2.2
Let
(Ω
,
F
)
be a measurable space on which
(
μ
n
:
n
∈
N
)
is an
increasing sequence of measures. Show that a measure
μ
on
(Ω
,
F
)
is deﬁned
by the rule
A
∈ F ⇒
μ
(
A
) = lim
μ
n
(
A
)
.
3
Problem 3.1
Let
φ
:
R
2
→
R
be continuous and
(Ω
,
F
)
a measurable space.
Show that if
f, g
∈
m
(Ω
,
F
)
then
φ
◦
(
f, g
)
∈
m
(Ω
,
F
)
. [In particular,
f
+
g
and
fg
are measurable.]
Problem 3.2
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 Fall '09
 Robinson

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