Pi-system

Summary

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

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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

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  • 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

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A 𝜆-system on   is a set   of subsets of   satisfying

  •  
  • if   then  
  • if   is a sequence of (pairwise) disjoint subsets in   then  

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

The π-𝜆 theorem

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Let   be a 𝜆-system, and let    be a π-system contained in   The π-𝜆 theorem[1] states that the 𝜎-algebra   generated by   is contained in    

The π-𝜆 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 𝜎-finite measures.[2]

The π-𝜆 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 π-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 𝜆-system is often relatively easy. Despite the difference between the two theorems, the π-𝜆 theorem is sometimes referred to as the monotone class theorem.[1]

Example

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Let   be two measures on the 𝜎-algebra   and suppose that   is generated by a π-system   If

  1.   for all   and
  2.  

then   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 𝜎-algebra, and so the problem of equating measures would be completely hopeless without such a tool.

Idea of the proof[2] Define the collection of sets   By the first assumption,   and   agree on   and thus   By the second assumption,   and it can further be shown that   is a 𝜆-system. It follows from the π-𝜆 theorem that   and so   That is to say, the measures agree on  

π-Systems in probability

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π-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 π-𝜆 theorem was proven by the probabilist Eugene Dynkin. Standard measure theory texts typically prove the same results via monotone classes, rather than π-systems.

Equality in distribution

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The π-𝜆 theorem motivates the common definition of the probability distribution of a random variable   in terms of its cumulative distribution function. Recall that the cumulative distribution of a random variable is defined as   whereas the seemingly more general law of the variable is the probability measure   where   is the Borel 𝜎-algebra. The random variables   and   (on two possibly different probability spaces) are equal in distribution (or law), denoted by   if they have the same cumulative distribution functions; that is, if   The motivation for the definition stems from the observation that if   then that is exactly to say that   and   agree on the π-system   which generates   and so by the example above:  

A similar result holds for the joint distribution of a random vector. For example, suppose   and   are two random variables defined on the same probability space   with respectively generated π-systems   and   The joint cumulative distribution function of   is  

However,   and   Because   is a π-system generated by the random pair   the π-𝜆 theorem is used to show that the joint cumulative distribution function suffices to determine the joint law of   In other words,   and   have the same distribution if and only if they have the same joint cumulative distribution function.

In the theory of stochastic processes, two processes   are known to be equal in distribution if and only if they agree on all finite-dimensional distributions; that is, for all    

The proof of this is another application of the π-𝜆 theorem.[3]

Independent random variables

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The theory of π-system plays an important role in the probabilistic notion of independence. If   and   are two random variables defined on the same probability space   then the random variables are independent if and only if their π-systems   satisfy for all   and     which is to say that   are independent. This actually is a special case of the use of π-systems for determining the distribution of  

Example

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Let   where   are iid standard normal random variables. Define the radius and argument (arctan) variables  

Then   and   are independent random variables.

To prove this, it is sufficient to show that the π-systems   are independent: that is, for all   and    

Confirming that this is the case is an exercise in changing variables. Fix   and   then the probability can be expressed as an integral of the probability density function of    

See also

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Notes

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  1. ^ The nullary (0-ary) intersection of subsets of   is by convention equal to   which is not required to be an element of a π-system.

Citations

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  1. ^ a b Kallenberg, Foundations Of Modern Probability, p. 2
  2. ^ a b Durrett, Probability Theory and Examples, p. 404
  3. ^ Kallenberg, Foundations Of Modern Probability, p. 48

References

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  • Gut, Allan (2005). Probability: A Graduate Course. Springer Texts in Statistics. New York: Springer. doi:10.1007/b138932. ISBN 0-387-22833-0.
  • Williams, David (1991). Probability with Martingales. Cambridge University Press. ISBN 0-521-40605-6.
  • Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020.