In mathematics, a π-system (or pi-system) on a set
Ω
{\displaystyle \Omega }
is a collection
P
{\displaystyle P}
of certain subsets of
Ω
,
{\displaystyle \Omega ,}
such that
P
{\displaystyle P}
is non-empty.
If
A
,
B
∈
P
{\displaystyle A,B\in P}
then
A
∩
B
∈
P
.
{\displaystyle A\cap B\in P.}
That is,
P
{\displaystyle P}
is a non-empty family of subsets of
Ω
{\displaystyle \Omega }
that is closed under non-empty finite intersections.
The importance of π-systems arises from the fact that if two probability measures agree on a π-system, then they agree on the 𝜎-algebra generated by that π-system. Moreover, if other properties, such as equality of integrals, hold for the π-system, then they hold for the generated 𝜎-algebra as well. This is the case whenever the collection of subsets for which the property holds is a 𝜆-system. π-systems are also useful for checking independence of random variables.
This is desirable because in practice, π-systems are often simpler to work with than 𝜎-algebras. For example, it may be awkward to work with 𝜎-algebras generated by infinitely many sets
σ
(
E
1
,
E
2
,
…
)
.
{\displaystyle \sigma (E_{1},E_{2},\ldots ).}
So instead we may examine the union of all 𝜎-algebras generated by finitely many sets
⋃
n
σ
(
E
1
,
…
,
E
n
)
.
{\textstyle \bigcup _{n}\sigma (E_{1},\ldots ,E_{n}).}
This forms a π-system that generates the desired 𝜎-algebra. Another example is the collection of all intervals of the real line, along with the empty set, which is a π-system that generates the very important Borel 𝜎-algebra of subsets of the real line.
== Definitions ==
A π-system is a non-empty collection of sets
P
{\displaystyle P}
that is closed under non-empty finite intersections, which is equivalent to
P
{\displaystyle P}
containing the intersection of any two of its elements.
If every set in this π-system is a subset of
Ω
{\displaystyle \Omega }
then it is called a π-system on
Ω
.
{\displaystyle \Omega .}
For any non-empty family
Σ
{\displaystyle \Sigma }
of subsets of
Ω
,
{\displaystyle \Omega ,}
there exists a π-system
I
Σ
,
{\displaystyle {\mathcal {I}}_{\Sigma },}
called the π-system generated by
Σ
{\displaystyle {\boldsymbol {\varSigma }}}
, that is the unique smallest π-system of
Ω
{\displaystyle \Omega }
containing every element of
Σ
.
{\displaystyle \Sigma .}
It is equal to the intersection of all π-systems containing
Σ
,
{\displaystyle \Sigma ,}
and can be explicitly described as the set of all possible non-empty finite intersections of elements of
Σ
:
{\displaystyle \Sigma :}
{
E
1
∩
⋯
∩
E
n
:
1
≤
n
∈
N
and
E
1
,
…
,
E
n
∈
Σ
}
.
{\displaystyle \left\{E_{1}\cap \cdots \cap E_{n}~:~1\leq n\in \mathbb {N} {\text{ and }}E_{1},\ldots ,E_{n}\in \Sigma \right\}.}
A non-empty family of sets has the finite intersection property if and only if the π-system it generates does not contain the empty set as an element.
== Examples ==
For any real numbers
a
{\displaystyle a}
and
b
,
{\displaystyle b,}
the intervals
(
−
∞
,
a
]
{\displaystyle (-\infty ,a]}
form a π-system, and the intervals
(
a
,
b
]
{\displaystyle (a,b]}
form a π-system if the empty set is also included.
The topology (collection of open subsets) of any topological space is a π-system.
Every filter is a π-system. Every π-system that doesn't contain the empty set is a prefilter (also known as a filter base).
For any measurable function
f
:
Ω
→
R
,
{\displaystyle f:\Omega \to \mathbb {R} ,}
the set
I
f
=
{
f
−
1
(
(
−
∞
,
x
]
)
:
x
∈
R
}
{\displaystyle {\mathcal {I}}_{f}=\left\{f^{-1}((-\infty ,x]):x\in \mathbb {R} \right\}}
defines a π-system, and is called the π-system generated by
f
.
{\displaystyle f.}
(Alternatively,
{
f
−
1
(
(
a
,
b
]
)
:
a
,
b
∈
R
,
a
<
b
}
∪
{
∅
}
{\displaystyle \left\{f^{-1}((a,b]):a,b\in \mathbb {R} ,a
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