Stat 310A/Math 230A Theory of Probability
Practice Midterm Solutions
Andrea Montanari
October 28, 2010
Problem 1
Consider the measurable space (Ω
,
F
), with: Ω =
{
0
,
1
}
N
the set of (inﬁnite) binary sequences
ω
=
(
ω
1
,ω
2
,ω
3
,...
);
F
the
σ
algebra generated by cylindrical sets (equivalently the
σ
algebra generated by
sets of the type
A
i,x
=
{
ω
:
ω
i
=
x
}
for
i
∈
N
and
x
∈ {
0
,
1
}
).
(a)
Let
f
: Ω
→
[0
,
1] be deﬁned by
f
(
ω
)
≡
∞
X
n
=1
ω
n
2
n
.
(1)
for
ω
= (
ω
1
,ω
2
,...
). Is
f
measurable? Prove your answer. (As usual, assume [0
,
1] to be endowed with the
Borel
σ
algebra.)
Solution :
Yes,
f
is measurable. Indeed consider the following collection of intervals
P
=
{
B
k,n
= [
k/
2
n
,
(
k
+ 1)
/
2
n
) :
,n,k
∈
N
, n
≥
0
,
0
≤
k
≤
2
n

1
} ∪
n
{
1
}
o
.
(2)
This is a
π
system (indeed if
n
≥
m
,
B
k,n
∩
B
l,m
is either empty or equal to
B
k,n
). Further
σ
(
P
) =
B
[0
,
1)
.
Indeed any interval of the form [
k/
2
n
,l/
2
n
) is disjoint union of a ﬁnite colection of intervals in
P
. As a
consequence any interval (
a,b
), 0
≤
a < b
≤
1 is in
σ
(
P
) since we can ﬁnd monotone sequences
a
n
=
k
n
/
2
n
> a
,
b
n
=
l
n
/
2
n
< b
with
a
n
↓
a
,
b
n
↑
b
, whence (
a,b
) =
∪
n
≥
n
0
[
a
n
,b
n
). By taking union intevals in
P
, we conclude that all intervals of the form [0
,b
), (
a,
1] are in
σ
(
P
) as well. This completes the proof that
σ
(
P
) =
B
[0
,
1)
.
It is therefore suﬃcient to show that
f

1
(
B
k,n
)
∈ F
for
k,n
as above. If the binary expansion of
k
is
k
=
k
1
2
n

1
+
k
2
2
n

2
+
···
+
k
n
, it is elementary to see that
f

1
(
B
k,n
) =
n
ω
such that
ω
1
=
k
1
,...,ω
n
=
k
n
o
\
n
ω
= (
k
1
,k
2
,...,k
n
,
1
,
1
,
1
,
1
,...
)
o
,
(3)
which is the intersection of two measuraable sets.
(b)
Let
g
: [0
,
1]
→
Ω be deﬁned by
g
(
x
)
≡
(
g
1
(
x
)
,g
2
(
x
)
,g
3
(
x
)
,...
)
,
(4)
g
i
(
x
)
≡ bb
2
i
x
cc
mod 2
,
(5)
where
bb
a
cc ≡
max
{
n
∈
N
:
n < a
}
. Is
g
a measurable mapping? Prove your answer. (Assume [0
,
1] and Ω
to be endowed the same
σ
algebras as above.)
Solution :
Consider the collection of sets
P
=
{
C
n
(
u
n
1
) :
n
∈
N
,u
n
1
∈ {
0
,
1
}
n
} ⊆ F
deﬁned by
C
n
(
u
n
1
) =
{
ω
:
ω
1
=
u
1
,ω
2
=
u
2
,...,ω
n
=
u
n
}
.
(6)
1