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
)=
(
|ξ
)
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
.