�
Section
1
Probability
Spaces,
Properties
of
Probability.
A
pair
(Π
,
S
)
is
a
measurable
space
if
S
is
a
ε
algebra
of
subsets
of
Π.
A
collection
A
of
subsets
of
Π
is
an
algebra
(ring)
if:
1.
Π
⊂
A.
2.
C, B
⊂
A
=
≥
C
B, C
B
⊂
A.
⇐
⇒
3.
B
⊂
A
=
≥
Π
\
B
⊂
A.
4.
A
is
a
ε
algebra,
if
in
addition,
C
i
⊂
A,
�
i
∗
1 =
≥
C
i
⊂
A.
i
∗
1
(Π
,
S
,
P
)
is
a
probability
space
if
P
is
a
probability
measure
on
S
,
i.e.
1.
P
(Π)
=
1
.
2.
P
(
A
)
∗
0
, A
⊂ S
.
�
↓
�
�
�
↓
3.
P
is
countably
additive:
A
i
⊂ S
,
�
i
∗
1
, A
i
⇐
A
j
=
∞ �
i
◦
=
j
=
≥
P
A
i
=
P
(
A
i
)
.
i
=1
i
=1
An
equivalent
formulation
of
Property
3
is:
3
�
.
P
is
a
finitely
additive
measure
and
⎩
B
n
∪
B
n
+1
,
B
n
=
B
=
≥
P
(
B
)
=
lim
P
(
B
n
)
.
n
n
∗
1
Lemma
1
Properties
3
and
3
�
are
equivalent.
Proof.
1
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�
�
�
�
�
�
�
�
�
�
�
�
3 =
≥
3
�
:
Let
C
n
=
B
n
\
B
n
+1
,
then
B
n
=
B
�
�
�
C
k
�
 all
disjoint.
k
∗
n
By
3,
P
(
B
n
) =
P
(
B
) +
P
(
C
k
)
∃
P
(
B
)
when
n
∃ ↓
.
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 Spring '05
 Panchenko
 Statistics, Algebra, Sets, Probability, measure

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