A cylinder set , sometimes called fringe events , is a special set that is used in measure theory , a branch of mathematics . A special case of a cylinder set is a rectangular cylinder . Systems of cylinder sets are used to define product σ-algebras , which in turn form the basis for the definition of product dimensions and product models .
definition
An arbitrary index set , a basic set, is given
I.
{\ displaystyle I}
Ω
: =
∏
i
∈
I.
Ω
i
{\ displaystyle \ Omega: = \ prod _ {i \ in I} \ Omega _ {i}}
as well as the canonical projection for a subset
J
⊂
I.
{\ displaystyle J \ subset I}
π
J
:
Ω
→
∏
i
∈
J
Ω
i
,
π
J
(
ω
)
=
ω
|
J
{\ displaystyle \ pi _ {J}: \ Omega \ to \ prod _ {i \ in J} \ Omega _ {i}, \ quad \ pi _ {J} (\ omega) = \ omega | _ {J} }
,
where the restriction to the components is referred to in. Then a lot is called the form
ω
|
J
{\ displaystyle \ omega | _ {J}}
J
{\ displaystyle J}
π
J
-
1
(
M.
)
⊂
Ω
For
M.
∈
Ω
J
: =
∏
i
∈
J
Ω
i
{\ displaystyle \ pi _ {J} ^ {- 1} (M) \ subset \ Omega {\ text {for}} M \ in \ Omega _ {J}: = \ prod _ {i \ in J} \ Omega _ {i}}
a set of cylinders with a base .
J
{\ displaystyle J}
Derived terms
System of cylinder quantities
If a σ-algebra is given on the set , the set system is called
Ω
J
{\ displaystyle \ Omega _ {J}}
A.
J
{\ displaystyle {\ mathcal {A}} _ {J}}
Z
J
: =
{
π
J
-
1
(
A.
J
)
|
A.
J
∈
A.
J
}
{\ displaystyle {\ mathcal {Z}} _ {J}: = \ {\ pi _ {J} ^ {- 1} (A_ {J}) \, | \, A_ {J} \ in {\ mathcal { A}} _ {J} \}}
the quantity system of cylinder quantities.
Rectangular cylinder
An element of the σ-algebra can be as Cartesian product of sets in the σ-algebra to write, so
A.
J
{\ displaystyle {\ mathcal {A}} _ {J}}
A.
j
{\ displaystyle {\ mathcal {A}} _ {j}}
Ω
i
{\ displaystyle \ Omega _ {i}}
A.
J
=
∏
i
∈
J
A.
i
For
A.
i
∈
A.
i
{\ displaystyle A_ {J} = \ prod _ {i \ in J} A_ {i} {\ text {for}} A_ {i} \ in {\ mathcal {A}} _ {i}}
,
this is what one calls a rectangular cylinder with a base . Then you define
A.
J
{\ displaystyle A_ {J}}
J
{\ displaystyle J}
Z
J
R.
: =
{
π
J
-
1
(
A.
J
)
|
A.
J
is a rectangular cylinder
}
{\ displaystyle {\ mathcal {Z}} _ {J} ^ {R}: = \ {\ pi _ {J} ^ {- 1} (A_ {J}) \, | \, A_ {J} {\ text {is a rectangular cylinder}} \}}
as a quantity system of all rectangular cylinders.
properties
One defines the set system
Z
: =
⋃
J
⊆
I.
J
at last
Z
J
{\ displaystyle {\ mathcal {Z}}: = \ bigcup _ {J \ subseteq I \ atop J {\ text {finite}}} {\ mathcal {Z}} _ {J}}
,
so this is a producer of the product σ-algebra of , so it's
A.
i
{\ displaystyle {\ mathcal {A}} _ {i}}
⨂
i
∈
I.
A.
i
=
σ
(
Z
)
{\ displaystyle \ bigotimes _ {i \ in I} {\ mathcal {A}} _ {i} = \ sigma ({\ mathcal {Z}})}
.
Likewise, the system of sets that arises from the union of all finite rectangular cylinders is
Z
R.
: =
⋃
J
⊆
I.
J
at last
Z
J
R.
{\ displaystyle {\ mathcal {Z}} ^ {R}: = \ bigcup _ {J \ subseteq I \ atop J {\ text {finite}}} {\ mathcal {Z}} _ {J} ^ {R} }
a producer of the product σ-algebra , so it is
A.
i
{\ displaystyle {\ mathcal {A}} _ {i}}
⨂
i
∈
I.
A.
i
=
σ
(
Z
R.
)
{\ displaystyle \ bigotimes _ {i \ in I} {\ mathcal {A}} _ {i} = \ sigma ({\ mathcal {Z}} ^ {R})}
.
literature
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