82
CHAPTER 4.
EXERCISES
7. Let
f
: Ω
→
R
be a map. Show that for
A
1
, A
2
,
· · · ⊆
R
it holds
f

1
∞
i
=1
A
i
=
∞
i
=1
f

1
(
A
i
)
.
8. Let (Ω
1
,
F
1
), (Ω
2
,
F
2
), (Ω
3
,
F
3
) be measurable spaces.
Assume that
f
: Ω
1
→
Ω
2
is (
F
1
,
F
2
)measurable and that
g
: Ω
2
→
Ω
3
is (
F
2
,
F
3
)
measurable. Show that then
g
◦
f
: Ω
1
→
Ω
3
defined by
(
g
◦
f
)(
ω
1
) :=
g
(
f
(
ω
1
))
is (
F
1
,
F
3
)measurable.
9. Prove Proposition 2.1.4.
10. Let (Ω
,
F
,
P
) be a probability space, (
M,
Σ) a measurable space and
assume that
f
: Ω
→
M
is (
F
,
Σ)measurable. Show that
μ
with
μ
(
B
) :=
P
(
{
ω
:
f
(
ω
)
∈
B
}
)
for
B
∈
Σ
is a probability measure on Σ.
11. Assume the probability space (Ω
,
F
,
P
) and let
f, g
: Ω
→
R
be mea
surable stepfunctions. Show that
f
and
g
are independent if and only
if for all
x, y
∈
R
it holds
P
(
{
f
=
x, g
=
y
}
) =
P
(
{
f
=
x
}
)
P
(
{
g
=
y
}
)
12. Assume the product space ([0
,
1]
×
[0
,
1]
,
B
([0
,
1])
⊗B
([0
,
1])
, λ
×
λ
) and
the random variables
f
(
x, y
) := 1I
[0
,p
)
(
x
) and
g
(
x, y
) := 1I
[0
,p
)
(
y
), where
0
< p <
1. Show that
(a)
f
and
g
are independent,
(b) the law of
f
+
g
,
P
f
+
g
(
{
k
}
)
,
for
k
= 0
,
1
,
2 is the binomial distri
bution
μ
2
,p
.