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for any non-negative measurable
f
:
Ω
→
R
. Therefore
(
)
−
(
)=
− α
+
α
Ω
(
ν
A
− μ
B
)
≥
m
B
m
A
fdP
fdQ
dQ
0.
Ω
Ω
Finally let
A
n
=(
μ
A
n
,
ν
A
n
)
∈M
,
A
=(
μ
A
,
ν
A
)
∈M
,
A
n
A
, i.e.
μ
A
n
μ
A
,
ν
A
n
ν
A
.We
have
m
(
A
n
)=
Ω
μ
A
n
dP
−
α
Ω
μ
A
n
dQ
+
α
−
α
Ω
ν
A
n
dQ
Ω
μ
A
dP
−
α
Ω
μ
A
dQ
+
α
−
α
Ω
ν
A
dQ
=
m
(
A
)
.
4. Observables
In the classical probability there are three main notions:
probability = measure
random variable = measurable function
mean value = integral.
The first notion has been studied in the previous section. Now we shall define the second two
notions.
ξ
−
1
(Ω
S
)
→
(
)
∈S
Classically a random variable is such function
ξ
:
,
R
that
A
for any Borel
set
A
∈B
(
R
)
(here
B
(
R
)=
σ
(
J
)
is the
σ
-algebra generated by the family
J
of all intervals).
Now instead of a
σ
-algebra
S
we have the family
F
of all IF-events, hence we must give to
any Borel set
A
an element of
. Of course, instead of random variable we shall use the term
observable ([15], [16], [18], [32], [35]).
F
Definition 3.1.
An observable is a mapping
σ
(
J
)
→F
x
:
satisfying the following conditions:
(i)
(
)=(
)
(∅)=(
)
x
R
1, 0
,
x
0, 1
,
(ii)
A
∩
B
=
∅
=
⇒
x
(
A
)
x
(
B
)=(
0, 1
)
,
x
(
A
∪
B
)=
x
(
A
)
⊕
x
(
B
)
,
(iii)
=
⇒
(
)
(
)
A
n
A
x
A
n
x
A
.
σ
(
J
)
→F
F→
[
]
Proposition 3.1.
If
x
:
is an observable, and
m
:
0, 1
is a state, then
=
◦
σ
(
J
)
→
[
]
m
x
m
x
:
0, 1
defined by
(
)=
(
(
))
m
x
A
m
x
A
is a probability measure.
