1.3. The Probability Set Function 19
Theorem 1.3.6.Let{Cn}be a nondecreasing sequence of events. Then
lim
n→∞
P(Cn)=P( lim
n→∞
Cn)=P
(∞
⋃
n=1
Cn
)
. (1.3.8)
Let{Cn}be a decreasing sequence of events. Then
lim
n→∞
P(Cn)=P( lim
n→∞
Cn)=P
(∞
⋂
n=1
Cn
)
. (1.3.9)
Proof. We prove the result (1.3.8) and leave the second result as Exercise 1.3.20.
Define the sets, called rings, asR 1 =C 1 and, forn>1,Rn=Cn∩Cnc− 1 .It
follows that
⋃∞
n=1Cn=
⋃∞
n=1Rnand thatRm∩Rn =φ,form^ = n.Also,
P(Rn)=P(Cn)−P(Cn− 1 ). Applying the third axiom of probability yields the
following string of equalities:
P
[
lim
n→∞
Cn
]
= P
(∞
⋃
n=1
Cn
)
=P
(∞
⋃
n=1
Rn
)
=
∑∞
n=1
P(Rn) = lim
n→∞
∑n
j=1
P(Rj)
= lim
n→∞
⎧
⎨
⎩
P(C 1 )+
∑n
j=2
[P(Cj)−P(Cj− 1 )]
⎫
⎬
⎭
= lim
n→∞
P(Cn).(1.3.10)
This is the desired result.
Another useful result for arbitrary unions is given by
Theorem 1.3.7(Boole’s Inequality).Let{Cn}be an arbitrary sequence of events.
Then
P
(∞
⋃
n=1
Cn
)
≤
∑∞
n=1
P(Cn). (1.3.11)
Proof: LetDn=
⋃n
i=1Ci.Then{Dn}is an increasing sequence of events that go
up to
⋃∞
n=1Cn. Also, for allj,Dj=Dj−^1 ∪Cj. Hence, by Theorem 1.3.5,
P(Dj)≤P(Dj− 1 )+P(Cj),
that is,
P(Dj)−P(Dj− 1 )≤P(Cj).
In this case, theCis are replaced by theDis in expression (1.3.10). Hence, using
the above inequality in this expression and the fact thatP(C 1 )=P(D 1 ), we have
P
(∞
⋃
n=1
Cn
)
= P
(∞
⋃
n=1
Dn
)
= lim
n→∞
⎧
⎨
⎩
P(D 1 )+
∑n
j=2
[P(Dj)−P(Dj− 1 )]
⎫
⎬
⎭
≤ lim
n→∞
∑n
j=1
P(Cj)=
∑∞
n=1
P(Cn).