Mathematical Preliminaries - Semantics of Probabilistic Processes: An Operational Approach - page 17

which is a contradiction. So x ∗ is a fixed point of f . It is also unique. Otherwise,
suppose there is another fixed point x ;wehave d ( x , x ∗ ) > 0 since x = x ∗ . But
then we would have

d ( x , x ∗ ) d ( f ( x ), f ( x ∗ )) d ( x , x ∗ ) <d ( x , x ∗ ) .

= ≤ ʴ ·

Therefore x ∗ is the unique fixed point of f .

2.5 Probability Spaces

In this section, we recall some basic concepts from probability and measure theory.
More details can be found in many excellent textbooks, for example [ 4 ].

Definition 2.19 Let X be an arbitrary nonempty set and X a collection of subsets

of X . We say that X is a field on X if

1. The empty set ∅∈ X ;

2. Whenever A ∈ X , then the complement X \ A ∈ X ;

∈ X ∪ ∈ X

3. Whenever A , B , then the union A B .

A field X is a ˃ -algebra , if it is closed under countable union: whenever A i ∈ X for

, then i ∈N A i ∈ X i ∈ N .

The elements of a ˃ -algebra are called measurable sets , and ( X , X ) is called a

measurable space . A measurable space ( X , X ) is called discrete if X is the powerset

P ( X ). A ˃ -algebra generated by a family of sets X , denoted ˃ ( X ), is the small-

est ˃ -algebra that contains X . The existence of ˃ ( X ) is ensured by the following

proposition.

Proposition 2.3 For any nonempty set X and X a collection of subsets of X, there

exists a unique smallest ˃ -algebra containing X .

The Borel ˃ -algebra on a topological space ( X , X ) is the smallest ˃ -algebra

containing . The elements of the Borel ˃ -algebra are called Borel sets .Ifwehave

a topological space then we can always consider its Borel ˃ -algebra and regard
( X , ˃ (

X

X )) as a measurable space.

Let X be a collection of subsets of a set X . We say X is a ˀ -class if it is closed

under finite intersections; X is a ʻ -class if it is closed under complementations and

countable disjoint unions.

Theorem 2.9 (The ˀ - ʻ theorem) If X is a ˀ -class, then ˃ ( X ) is the smallest ʻ-class

containing X .

Definition 2.20 Let ( X , X ) be a measurable space. A function μ : X ₒ [ −∞ , ∞ ]

X

is a measure on if it satisfies the following conditions:

≥ ∈ X

1. μ ( A ) 0 for all A ;

2. μ ( 0;

3. If A 1 , A 2 , ... are in

∅ ) =

for i = j , then μ ( i A i ) = i μ ( A i ). X , with A i ∩ A j =∅