Continuity Property of Probability
Definition: A sequence of events <En> is said to be an increasing sequence if
E1  E2  E3  …. A sequence of events <En> is said to be a decreasing sequence if
E1  E2  E3  …. If <En> is an increasing sequence of events, we define a new event, denoted

by lim E n , by lim E n   E n . If <En> is a decreasing sequence of events, we define a new
n 
n 
n 1

event, denoted by lim E n , by lim E n   E n .
n 

n 
n 1

Proposition: If <En> is either an increasing or decreasing sequence of events, then
lim PE n   P lim E n .
n 
n 
Proof: First, suppose that <En> is an increasing sequence of events, and define a new sequence
C
 n 1 
of events <Fn> by: i) F1 = E1, and ii) Fn  E n    Ei   E n  E nC1 , for n > 1. Then <Fn> is
 i 1 
an increasing sequence of mutually exclusive events such that


i 1
i 1
 Fi   Ei , and
n
n
i 1
i 1
 Fi   Ei ,
for all n  1. Thus
n
 
  
 n 
 n

P  Fi   P  Ei    PFi   lim  PFi   lim P  Fi   lim P  Ei   lim PE n ,
n 
n 
i 1
 i 1 
 i 1  i 1
 i 1  n  i 1  n
proving the result when the sequence is increasing.
Now suppose that <En> is a decreasing sequence of events. Then  E nC  is an increasing


sequence of events. From the preceding equations, we have P  E nC   lim P E nC . But since
 n 1
 n 
C
C

 


 
C

E n    E n  , we see that P   E n   lim P E nC , or equivalently

  n1   n
n 1
 n 1 








1  P  E n   lim 1  PE n  , or P  E n   lim PE n  , so that P lim E n  lim PE n  , proving
n 
n 
 n 1  n
 n 1  n
the result when the sequence is decreasing.
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 
 

