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In mathematics, a π-system (or pi-system) on a set is a collection of certain subsets of such that
- is non-empty.
- If then
That is, is a non-empty family of subsets of that is closed under non-empty finite intersections.[nb 1] 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 So instead we may examine the union of all 𝜎-algebras generated by finitely many sets 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 that is closed under non-empty finite intersections, which is equivalent to containing the intersection of any two of its elements. If every set in this π-system is a subset of then it is called a π-system on
For any non-empty family of subsets of there exists a π-system called the π-system generated by , that is the unique smallest π-system of containing every element of It is equal to the intersection of all π-systems containing and can be explicitly described as the set of all possible non-empty finite intersections of elements of
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 and the intervals form a π-system, and the intervals 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 the set defines a π-system, and is called the π-system generated by (Alternatively, defines a π-system generated by )
- If and are π-systems for and respectively, then is a π-system for the Cartesian product
- Every 𝜎-algebra is a π-system.
Relationship to 𝜆-systems
A 𝜆-system on is a set of subsets of satisfying
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