Lecture Notes in Computer Science 3472

236 Verena Wolf
• A finite trace β is a sequence
β = a0 a1 ...an−1 ∈ Act ∗ .
Let trace(α) denote the ordered sequence of all external actions occurring in
a finite path α. Note that finite paths always end up in a terminal state and
start in the initial state whereas traces can be arbitrary sequences of Act ∗ .
• We use the previous definitions of terminal states, τ-free, finitely-branching
and divergence-free non-probabilistic processes also for all the probabilistic
extensions that are defined later (the formal definitions are analogous). We
will use the term process for all kinds of processes considered in the sequel
and from now on all processes are divergence-free and finitely-branching.
Weight functions: Aweight function on a countable set S is a function δ :
S → R�0. LetWeight(S) denote the set of all weight functions on S.
Distributions:
�
µ ∈ Weight(S) is a distribution on a countable set S if
s∈S µ(s) =1. Let supp(µ) ={s ∈ S : µ(s) > 0} and Distr(S) betheset
of all distributions on S. Ifs ∈ S then χs denotes the unique distribution on S
with
�
1 if t = s,
χs(t) =
0 if t ∈ S \{s}.
The product µ × λ of two distributions µ on a set S and λ on a set T is the
distribution on S × T defined by
(µ × λ)(s, t) =µ(s) · λ(t), s ∈ S, t ∈ T .
In the following λ, µ, π, σ will always denote distributions.
Probability spaces: A probability space is a tuple (Ω,A, P), where
• Ω is a nonempty set of outcomes,
•A⊆P(Ω) isaσ-algebra, i.e. Ω ∈Aand A is closed under countable union
and complement,
• P : A→[0, 1] is a probability measure, i.e. P(Ω) =1,P(∅) =0andfor
countably many pairwise disjoint A1, A2,...∈Awe have
�
i P(Ai) =P( �
i Ai).
For C⊆P(Ω) letσ(C) denote the smallest σ-algebra containing C defined by
�
σ(C) =
A.
C⊂A
A is σ-algebra on Ω
A probability measure defined on an appropriate set C canbeextendedtoa
unique probability measure on σ(C) (for details we refer to the book of Feller
[Fel68]).
Example. Consider Ω =[0, 1] and C = {]a, b] | 0 � a < b � 1}. Thenσ(C) is
the set of all countable unions of closed or open subsets of [0, 1] and P(]a, b]) =
b − a can be extended to a unique probability measure (the so-called Lebesgue
measure) on σ(C).
⊓⊔