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ECON 831
Solution Homework #4
1. Bierens, 11
Consider
F
an algebra.
(i) First, we show that
F
σ
algebra
⇒ F
monotone class.
By de nition of a
σ
algebra, we know that for any sequence of sets in
F
, their in nite
union is also an element of
F
, and their in nite intersection is also an element of
F
. This
is true in particular for increasing and decreasing sequences. Hence the two properties
characterizing a monotone class are satis ed, and we conclude that
F
is a monotone
class.
(ii) Second, we show that
F
monotone class
⇒ F
σ
algebra.
By assumption
F
is an algebra, and we need to show that it is also a
σ
algebra. That
is, we need to show that for any sequence of sets in
F
, their in nite union is still an
element of
F
. Consider then an arbitrary sequence of sets in
F
, say
A
n
∈ F
. We now
construct a new sequence,
C
n
, s.t.
C
i
=
S
i
k
=1
A
k
: so we have
C
1
=
A
1
,
C
2
=
A
1
∪
A
2
,
... By de nition, the sequence
C
n
is increasing, that is
C
n
⊂
C
n
+1
,
C
n
∈ F
, and also
S
i
k
=1
A
k
=
S
i
k
=1
C
k
. By assumption we know that
F
is a monotone class, so we can
use one of its property to deduce that
S
∞
k
=1
C
k
∈ F
. And since
S
∞
k
=1
A
k
=
S
∞
k
=1
C
k
, we
have
S
∞
k
=1
A
k
∈ F
. We conclude that
F
is a
σ
algebra.
2. Bierens, 12
Consider
F
a
λ
system. We show that
F
π
system
⇒ F
σ
algebra. We need to check
the three properties that de ne a
σ
algebra. First, let me recall the three properties
satisfy by a
πλ
system:
(1)
A
∈ F ⇒
A
c
∈ F
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 Fall '09
 Antoine
 Economics

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