Page 17 - Elements of Distribution Theory

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1.2 Basic Framework 3

d
d
Euclidean space R , F may be taken to be the set of all subsets of R formed by starting
d
with a countable set of rectangles in R and then performing a countable number of set
operations such as intersections and unions. The same approach works when is a subset
of a Euclidean space.
The study of theses issues forms the branch of mathematics known as measure theory.
In this book, we avoid such issues and implicitly assume that any event of interest is
measurable.
Note that condition (P3), which deals with an infinite number of events, is of a different
nature than conditions (P1) and (P2). This condition is often referred to as countable additiv-
ity of a probability function. However, it is best understood as a type of continuity condition
on P. It is easier to see the connection between (P3) and continuity if it is expressed in terms
of one of two equivalent conditions. Consider the following:
(P4) If A 1 , A 2 ,..., are subsets of satisfying A 1 ⊂ A 2 ⊂· · · , then

∞

P A n = lim P(A n )
n→∞
n=1
(P5) If A 1 , A 2 ,..., are subsets of satisfying A 1 ⊃ A 2 ⊃· · · , then

∞

P A n = lim P(A n ).
n→∞
n=1
Suppose that, as in (P4), A 1 , A 2 ,... is a sequence of increasing subsets of . Then we
may take the limit of this sequence to be the union of the A n ; that is,
∞

lim A n = A n .
n→∞
n=1
Condition (P4) may then be written as

P lim A n = lim P(A n ).
n→∞ n→∞
A similar interpretation applies to (P5). Thus, (P4) and (P5) may be viewed as continuity
conditions on P.
The equivalence of (P3), (P4), and (P5) is established in the following theorem.

Theorem 1.1. Consider an experiment with sample space . Let P denote a function defined
on subsets of such that conditions (P1) and (P2) are satisfied. Then conditions (P3), (P4),
and (P5) are equivalent in the sense that if any one of these conditions holds, the other two
hold as well.

Proof. First note that if A 1 , A 2 ,... is an increasing sequence of subsets of , then
c
c
A , A ,... is a decreasing sequence of subsets and, since, for each k = 1, 2,...,
1 2
c
k k

c
A n = A ,
n
n=1 n=1
c ∞
c c
lim A n = A = lim A .
n
n
n→∞ n→∞
n=1