General Probability Spaces
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71
General Probability Spaces
The section deals with the general probability space and the
measurable space.To this end we introduce rst the notion
of the σ eld; important is the Borel σ eld over Rn as a
special case.
In section 3 the probability measure was introduced
over a discrete probability space, what means a considerable simplication. As system of events (totality
of all possible events) we could use the power set
P(Ω) (set of all subsets of a given set) of the discrete basic space Ω. A concept which in the case
of over-countable basic spaces has to be given up
by reasons of logical contradictions. For evidence
we mention, that there does not exist a translation
invariant measure on the power set P(R) of R; i.e.
there does not exist a measure on P(R) which assigns
the same measure value to sets generated by translation of a given set.
A way-out consists in dening measures on σ -elds;
in the case of Rn , especially the Borel σ eld will be
choosen. For the treatment of practical problems in
the Rn there result almost no limitations.
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7.1 Denition
A system A of subsets over Ω is called σ eld (over
Ω), if:
(7.1.1)
Ω∈A
(7.1.2)
A ∈ A ⇒ Ac ∈ A
(7.1.3)
with a sequence (An ) of sets of A we always have
∞
[
An ∈ A .
n=1
Examples of σ elds over Ω6= ∅ are e.g. the systems
{∅, Ω} or {∅, A, Ac , Ω} for A ⊂ Ω or P(Ω) .
As an immediate consequence of 7.1 we have the
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7.2 Conclusions
7.2.1 ∅ ∈ A .
7.2.2 If (An ) is a sequence of sets of A, we also have
∞
\
An ∈ A .
7.2.3 Ai ∈ A, i = 1, . . .n=1
,n ⇒
n
[
i=1
Ai ∈ A,
n
\
Ai ∈ A .
i=1
7.2.4 If Ai ∈ A, i = 1, 2 we also have A1 \ A2 ∈ A .
For 7.2.2 and 7.2.3 resp. one also is saying, that a σ eld is closed for operations of forming unions
and intersections of nite or countable many
elements of the σ eld.
7.3 Borel σ eld
Over R, Rn the Borel σ eld B B n will be used as
the standard σ eld resp.; on a formal introduction
will be renounced here.
The σ elds B and B n resp. satisfy the conditions
(7.1.1) (7.1.3).
Moreover B contains the set systems (set of sets)
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• of all halfintervals (−∞, x] with x ∈ R
• of all intervals (a, b), (a, b], [a, b) and [a, b] with
a≤b⊂R
• of all open sets of R.
A corresponding fact can be formulated for B n .
Note, Bn contains also all singletons, i.e. all {x} ⊂
Rn mit x ∈ Rn ; a fact which for arbitrary σ elds need
not be true; cf. e.g. the σ -eld {∅, Ω} over Ω.
As the Borel σ eld B n is used as the standard
space of denition for probability measures over
Rn we have that also the singleton point sets of Rn
probability values are assigned.
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7.4 Denition (general probability space, measure space)
7.4.1 The tripel (Ω, A, P ) with Ω as nonempty
set, A as σ eld over Ω and a mapping
P : A → R+
(7.4.1.1)with
P (Ω) = 1
(7.4.1.2)
P (A) ≥ 0
(normalization),
(nonnegativity),
For any sequence (An ) mutually disjoint sets A we
have
(7.4.1.3)
!
∞
∞
X
X
P
An =
P (An ) (σ − additivity)
1
1
is called (general) probability space); Ω is called basic space; A stands for the system of events; while
P denotes probability measure. The elements of
A are called measurable sets. (Ω, A) is called measurable space; we are speaking of P as the probability
measure over the measurable space (Ω, A).
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7.4.2 If the condition (7.4.1.1) P (Ω) = 1 are replaced by P (∅) = 0, one speaks of P as of a
measure denoting it now by µ, ν instead of P
etc.; (Ω, A, µ) is called measure space.
∞
P
( An denotes the disjoint union, i.e. the
1
union of the disjoint sets An )
Note: Probabilities are assigned only to the
measurable sets, i.e. the elements of the σ eld
A.
7.5 The realisations according to a probability
measure (probabilitylaw)
In the intuitive, outer-mathematical interpretation P (A) stands for the probability, that an ω̄ ∈ Ω,
randomly realized in accordance with P , (established by measurement or observation), falls into the
set A, i.e., that ω̄ ∈ A for A ∈ A holds.
One speaks of ω̄ as of a realisation according to
the (probability law) P .
The empirical examination, if given realisation
are those according to P , is only possible for a
whole collection of outcomes; cf. Experiment 15.1 .
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7.6 BorelLebesgue measure
7.6.1 Over (R, B) a measure µ is without proof uniquely determined through
µ((a, b]) = b − a
for a < b ∈ R
µ is called the BorelLebesgue measure (BL
measure) over B, where instead of µ especially the symbol λ is used.
7.6.2 λ represents a general measure on B over
R; λ is not a probability measure.
If now A ∈ B is a subset of R, i.e. the interval [0; 1], the so-called restriction λA of λ
on A represents a probability measure; one
speaks of the uniform distribution over A
((general) uniform distribution)
λA (B) :=
λ(B)
λ(A)
(B ⊂ A, B ∈ B) .
Obviously we have λA (A) = 1 .
For A := [0; 1] we have
λ[0;1] (B) = λ(B)
(B ⊂ A, B ∈ B) .