Continuity Property of Probability
Continuity Property of Probability
Continuity Property of Probability
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 E i E n E nC1 , for n > 1. Then <Fn> is an
i 1
n n
all n 1. Thus
n
n
n
P F i P E i P Fi lim P Fi lim P F i lim P E i lim P E n ,
i 1 i 1 i 1
n
i 1
n
i 1 n
i 1 n
C
C
, or equivalently
E C
E , we see that P E n lim P E nC
n n n
n 1 n 1 n 1
1 P E n lim 1 P E n , or P E n lim P E n , so that
n 1 n
n 1 n