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ECE 514
, Fall 2007
Exam 1: Due 12:30pm in class, September 27, 2007
Solutions
(version: September 25, 2007, 10:55)
75 mins.; Total 50 pts.
1.
(12 pts.)
a. Consider a sample space
Ω
. Suppose
C
1
and
C
2
are collections of subsets of
Ω
such that
C
1
⊂ C
2
. Show that
σ
(
C
1
)
⊂
σ
(
C
2
)
.
b. Let
C
be the collection of all closed intervals on
R
of the form
[
a, b
]
where
a < b
,
a, b
∈
R
.
Is
σ
(
C
)
equal to the Borel
σ
algebra? Justify your answer fully.
Ans.:
a. We ﬁrst claim that
C
1
⊂
σ
(
C
2
)
. This follows from the fact that
C
1
⊂ C
2
(given) and that
C
2
⊂
σ
(
C
2
)
(by deﬁnition of
σ
(
C
2
)
).
With the claim above, we can now show the desired result. First, notice that
σ
(
C
2
)
is a
σ
algebra
(by deﬁnition), and it also contains
C
1
(by argument above). Hence, because
σ
(
C
1
)
is the smallest
such
σ
algebra,
σ
(
C
1
)
must be smaller than
σ
(
C
2
)
. This means that
σ
(
C
1
)
⊂
σ
(
C
2
)
.
b. The answer is yes. Let
C
2
be the collection of all intervals on
R
. Now,
σ
(
C
2
)
is the Borel
σ
algebra, by deﬁnition. Because
C ⊂ C
2
, by part a we have
σ
(
C
)
⊂
σ
(
C
2
)
. So it remains to show
that
σ
(
C
2
)
⊂
σ
(
C
)
. To show this, it sufﬁces to show that
σ
(
C
)
is a
σ
algebra that contains
C
2
(the
desired result then follows because
σ
(
C
2
)
is the smallest such
σ
algebra). First, it is clear that
σ
(
C
)
is a
σ
algebra. To show that it contains
C
2
, we let
A
∈ C
2
and show that
A
∈
σ
(
C
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