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Typically we will take
F
to be all subsets of
S
, and so (i)(iii) are automatically satisﬁed. The only
times we won’t have
F
be all subsets is for technical reasons or when we talk about conditional expectation.
So now we have a space
S
, a
σ
ﬁeld
F
, and we need to talk about what a probability is. There are
three axioms:
(1)
0
≤
P
(
E
)
≤
1
for all events
E
.
(2)
P
(
S
) = 1
.
(3) If the
E
i
are pairwise disjoint,
P
(
∪
∞
i
=1
E
i
) =
∑
∞
i
=1
P
(
E
i
)
.
Pairwise disjoint means that
E
i
∩
E
j
=
∅
unless
i
=
j
.
Note that probabilities are probabilities of subsets of
S
, not of points of
S
. However it is common to
write
P
(
x
) for
P
(
{
x
}
).
Intuitively, the probability of
E
should be the number of times
E
occurs in
n
times, taking a limit as
n
tends to inﬁnity. This is hard to use. It is better to start with these axioms, and then to prove that the
probability of
E
is the limit as we hoped.
There are some easy consequences of the axioms.
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This note was uploaded on 12/29/2011 for the course MATH 317 taught by Professor Wen during the Spring '09 term at SUNY Stony Brook.
 Spring '09
 wen
 Sets

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