Information Technology Reference
In-Depth Information
The
number P
(
A
|
B
)
can
be
regarded
as
a
constant
function,
Constant
functions
are
σ
S
= {
Ω }
measurable with respect to the
-algebra
,
.
0
(
|S
)
S
⊂S
S
Generally P
A
can be defined for any
σ
-algebra
,asan
0 -measurable function
0
0
such that
P
(
A
C
)=
P
(
A
|S 0 )
dP , C
∈S o .
C
S
= S
(
|S
)= χ A , since
χ A is
S
If
, then we can put P
A
0 -measurable, and
0
0
=
(
)
C χ A dP
P
A
C
.
An important example of
S 0 is the family of all pre-images of a random variable
ξ
:
Ω
R
S 0 = { ξ 1
(
B
)
; B
σ ( J ) }
.
(
|S
)=
(
)
In this case we shall write P
A
P
A
, hence
0
= ξ 1
C (
P
(
A
| ξ )
dP
=
P
(
A
C
)
, C
(
B
)
, B
σ ( J )
.
By the transformation formula
∩ ξ 1
(
(
)) =
◦ ξ
=
∈ σ ( J )
P
A
B
g
dP
gdP
, B
ξ
ξ 1
(
)
B
B
And exactly this formulation will be used in our IF -case,
m
(
A . x
(
B
)) =
p
(
A
|
x
)
dm x
=
p
(
A
|
x
)
dF .
B
B
(
|
)
Of course, we must first prove the existence of such a mapping p
A
x
: R
R ([34], [70],
[72]). Recall that the product of IF -events is defined by the formula
=( μ K .
ν K + ν L − ν K .
ν L )
K . L
μ L ,
.
σ ( J ) →F
F→ [
]
∈F
Theorem 5.1.
Let x :
be an observable, m :
0, 1
be a state, and let A
.
σ ( J ) [
]
Define
ν
:
0, 1
by the equality
ν (
)=
(
(
))
B
m
A . x
B
.
Then
ν
is a measure.
= ∅
∈B (
)= σ ( J )
(
)
(
)=(
)
Proof. Let B
C
, B , C
R
. Then x
B
. x
C
0, 1
, hence
A .
(
x
(
B
)
x
(
C
)) = (
A . x
(
B
)) (
A . x
(
C
))
,
and therefore
ν (
B
C
)=
m
(
A . x
(
B
C
)) =
m
(
A .
(
x
(
B
)
x
(
C
)) =
m
((
A . x
(
B
)) (
A . x
(
C
))) =
=
(
(
)) +
(
(
)) = ν (
)+ ν (
)
m
A . x
B
m
A . x
C
B
C
.
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