Mathematics 741  Advanced Probability Theory I  Fall 2008  Zirbel
Exercises to accompany the lecture notes
These exercises may or may not be assigned as homework for Math 741. Some of them
are referred to in the lecture notes by number.
Let (Ω
,
H
,
IP) be a probability space.
1.
a) Let
A
1
⊆
A
2
⊆ · · ·
be sets in
H
. Let
A
=
∞
∪
i
=1
A
i
. Show that IP(
A
i
) increases to
IP(
A
) as
i
→ ∞
.
b) Let
A
1
⊇
A
2
⊇ · · ·
be sets in
H
. Let
A
=
∞
∩
i
=1
A
i
. Show that IP(
A
i
) decreases to
IP(
A
) as
i
→ ∞
.
2.
Let
F
be the smallest
σ
algebra on
R
containing all intervals of the form (
−∞
, b
],
b
∈
R
. Show that
F
=
B
R
. We say that
B
R
is
generated
by the open subsets of
R
or
by intervals of the form (
−∞
, b
]. We write
B
R
=
σ
(
{
(
−∞
, b
] :
b
∈
R
}
).
Hint:
Show that
F ⊆ B
R
and then that
B
R
⊆ F
. Use the fact that
B
R
contains all
sets of the form (
−∞
, b
]. Show that
F
contains all open intervals in
R
and thus all
open subsets of
R
(each open subset is a countable union of open intervals).
3.
a) Let Ω and
S
be sets and let
S
be a
σ
algebra on
S
.
Let
X
: Ω
→
S
.
Define a
collection
σ
(
X
) of subsets of Ω by
σ
(
X
) =
{
X

1
(
B
) :
B
∈ S}
. Show that
σ
(
X
) is a
σ
algebra on Ω. Show all details.
b) Let
H
be a
σ
algebra on Ω. Let
D
=
{
B
⊆
S
:
X

1
(
B
)
∈ H}
. Show that
D
is
a
σ
algebra on
S
. Think of
D
as all subsets of
S
whose inverse images under
X
are
“nice”.
Notes:
(i)
σ
(
X
) is called the
σ
algebra generated by
X
.
It is the collection of all
events concerning
X
. (ii) The function
X
: (Ω
,
H
)
→
(
S,
S
) is measurable if and only
if
σ
(
X
)
⊆ H
. Think of
σ
(
X
) as the subsets of Ω that need to be measurable for the
mapping
X
to be measurable.
4.
Represent Ω by a square in the plane with sides of length 1. Divide Ω in half hor
izontally and vertically.
Let
X
: Ω
→
R
take on the values 1, 2, 3, 4 in the four
smaller squares. Represent
X
by showing the four values in the four smaller squares,
and represent subsets of Ω by shading in the appropriate squares. For example, the
set
{
X
= 1
}
is the upper–left smaller square.
Represent all elements of
σ
(
X
) by
shading
and in the form
X

1
(
B
) for different sets
B
. Check
σ
(
X
) to make sure it
is
a
σ
algebra.
5.
Let lim sup
n
→∞
A
n
≡
∞
∩
n
=1
∞
∪
m
=
n
A
m
.
Show that IP(lim sup
n
→∞
A
n
)
≥
lim sup
n
→∞
IP(
A
n
).
6.
a) Let
X
: Ω
→
S
.
Let
C
be a nonempty collection of subsets of
S
.
Show that
σ
(
X

1
(
C
)) =
X

1
(
σ
(
C
)). Here,
X

1
of a collection means the collection of inverse
images of each element of the collection. For example,
X

1
(
C
) =
{
X

1
(
B
) :
B
∈ C}
.
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 Spring '10
 Zirbel
 Probability theory, measure

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