Page 29 - Probability and Statistical Inference

6 1. Nations of Probability

sets A , ∈ I. Hence, the element x must belong to the set ∩ i∈ I (A ). Thus, we
c
i
i
have (∪ A ) ⊆ ∩ (A )
c
c
i∈ I i i∈ I i
Suppose that an element x belongs to the rhs of (1.2.5). That is x ∈ A for
c
i
each i ∈ I, which implies that x can not belong to any of the sets A , i ∈ I. In
i
other words, the element x can not belong to the set ∪ i∈ I A so that x must
i
belong to the set (∪ i∈ I A ) Thus, we have (∪ i∈ I A ) ⊇ ∩ (A ). The proof is
c
c
c
i
i
i
i∈ I
now complete. !
Definition 1.2.1 The collection of sets {A ; i ∈ I} is said to consist of
i
disjoint sets if and only if no two sets in this collection share a common ele-
ment, that is when A ∩ A = ϕ for all i ≠ j ∈ I. The collection {A ; ∈ I} is
i
i
j
called a partition of S if and only if
(i) {A ; i ∈ I} consists of disjoint sets only, and
i
(ii) {A ; i ∈ I} spans the whole space S, that is ∪ A = S.
i i∈ I i
Example 1.2.3 Let S = (0,1] and define the collection of sets {A ; i∈ I}
i
where A = ( i ∈ I = {1,2,3, ...}. One should check that the given
i
collection of intervals form a partition of (0,1]. !
1.3 Axiomatic Development of Probability
The axiomatic theory of probability was developed by Kolmogorov in his 1933
monograph, originally written in German. Its English translation is cited as
Kolmogorov (1950b). Before we describe this approach, we need to fix some
ideas first. Along the lines of the examples discussed in the Introduction, let us
focus on some random experiment in general and state a few definitions.
Definition 1.3.1 A sample space is a set, denoted by S, which enumerates
each and every possible outcome or simple event.
In general an event is an appropriate subset of the sample space S, includ-
ing the empty subset ϕ and the whole set S. In what follows we make this
notion more precise.
Definition 1.3.2 Suppose that ß = {A : A ⊆ S,i ∈ I} is a collection of
i i
subsets of S. Then, ß is called a Borel sigma-field or Borel sigma-algebra if
the following conditions hold:
(i) The empty set ϕ ∈ ß;
(ii) If A ∈ ß, then A ∈ ß;
c
(iii) If A ∈ ß for i = 1,2, ..., then ∈ ß.
i
In other words, the Borel sigma-field ß is closed under the operations of
complement and countable union of its members. It is obvious that the