In mathematics, a -system (or pi-system) on a set is a collection of certain subsets such that

• is non-empty.

that is closed under non-empty finite intersections. The importance of -systems arises from the fact that if two probability measures agree on a -system, then they agree on the -algebra generated by that -system. Moreover, if other properties, such as equality of integrals, hold for the -system, then they hold for the generated -algebra as well. This is the case whenever the collection of subsets for which the property holds is a -system. -systems are also useful for checking independence of random variables.

This is desirable because in practice, -systems are often simpler to work with than -algebras. For example, it may be awkward to work with So instead we may examine the union of all This forms a -system that generates the desired -algebra. Another example is the collection of all intervals of the real line, along with the empty set, which is a -system that generates the very important Borel -algebra of subsets of the real line.

Definitions

A -system containing the intersection of any two of its elements. If every set in this then it is called a

For any non-empty family there exists a called the , that is the unique smallest It is equal to the intersection of all

A non-empty family of sets has the finite intersection property if and only if the -system it generates does not contain the empty set as an element.

Examples

• form a form a -system if the empty set is also included.

• The topology (collection of open subsets) of any topological space is a -system.

• Every filter is a -system. Every -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 -system, and is called the -system defines a )

• are is a -system for the Cartesian product

• Every -algebra is a -system.

Relationship to -systems

A -system satisfying

• is a sequence of (pairwise) disjoint

Whilst it is true that any -algebra satisfies the properties of being both a -system and a -system, it is not true that any -system is a -system, and moreover it is not true that any -system is a -algebra. However, a useful classification is that any set system which is both a -system and a -system is a -algebra. This is used as a step in proving the - theorem.

The - theorem

be a -system, and let be a The - theorem states that the

The - theorem can be used to prove many elementary measure theoretic results. For instance, it is used in proving the uniqueness claim of the Carathéodory extension theorem for -finite measures.

The - theorem is closely related to the monotone class theorem, which provides a similar relationship between monotone classes and algebras, and can be used to derive many of the same results. Since -systems are simpler classes than algebras, it can be easier to identify the sets that are in them while, on the other hand, checking whether the property under consideration determines a -system is often relatively easy. Despite the difference between the two theorems, the - theorem is sometimes referred to as the monotone class theorem.

Example

be two measures on the is generated by a If

1. and

This is the uniqueness statement of the Carathéodory extension theorem for finite measures. If this result does not seem very remarkable, consider the fact that it usually is very difficult or even impossible to fully describe every set in the -algebra, and so the problem of equating measures would be completely hopeless without such a tool.

Idea of the proof is a -system. It follows from the -

-Systems in probability

-systems are more commonly used in the study of probability theory than in the general field of measure theory. This is primarily due to probabilistic notions such as independence, though it may also be a consequence of the fact that the - theorem was proven by the probabilist Eugene Dynkin. Standard measure theory texts typically prove the same results via monotone classes, rather than -systems.

Equality in distribution

<onlyinclude> The - theorem motivates the common definition of the probability distribution of a random variable in terms of its cumulative distribution function whereas the seemingly more general is the Borel (on two possibly different probability spaces) are (or agree on the and so by the example above

with respectively generated

is a the - have the same distribution if and only if they have the same joint cumulative distribution function.

In the theory of stochastic processes

The proof of this is another application of the - theorem.</onlyinclude>

Independent random variables

The theory of -system plays an important role in the probabilistic notion of independence then the random variables are independent if and only if their are independent. This actually is a special case of the use of

Example

are iid

are independent random variables.

To prove this, it is sufficient to show that the

See also

Notes

Citations

References