Pi-system

In mathematics, a pi-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. The importance of pi-systems arises from the fact that if two probability measures agree on a pi-system, then they agree on the sigma-algebra generated by that pi-system. Moreover, if other properties, such as equality of integrals, hold for the pi-system, then they hold for the generated sigma-algebra as well. This is the case whenever the collection of subsets for which the property holds is a lambda-system. pi-systems are also useful for checking independence of random variables. This is desirable because in practice, pi-systems are often simpler to work with than sigma-algebras. For example, it may be awkward to work with sigma-algebras generated by infinitely many sets So instead we may examine the union of all sigma-algebras generated by finitely many sets This forms a pi-system that generates the desired sigma-algebra. Another example is the collection of all intervals of the real line, along with the empty set, which is a pi-system that generates the very important Borel sigma-algebra of subsets of the real line. A pi-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 pi-system is a subset of then it is called a For any non-empty family of subsets of there exists a pi-system called the , that is the unique smallest pi-system of containing every element of It is equal to the intersection of all pi-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 pi-system it generates does not contain the empty set as an element. For any real numbers and the intervals form a pi-system, and the intervals form a pi-system if the empty set is also included.