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{{Short description|Family of sets closed under intersection}}
{{this|{{pi}}-system in mathematics|{{pi}}-systems in chemistry|pi bond}}
In [[Mathematics|mathematics]], a '''{{pi}}-system''' (or '''pi-system''') on a [[Set (mathematics)|set]] <math>\Omega</math> is a [[Family of sets|collection]] <math>P</math> of certain [[Subset|subset]]s of <math>\Omega,</math> such that

* <math>P</math> is [[Empty set|non-empty]].
* If <math>A, B \in P</math> then <math>A \cap B \in P.</math>

That is, <math>P</math> is a non-empty family of subsets of <math>\Omega</math> that is [[Closure (mathematics)|closed]] under non-empty finite [[Intersection (set theory)|intersections]].<ref group=nb>The nullary (0-ary) intersection of subsets of <math>\Omega</math> is by convention equal to <math>\Omega,</math> which is not required to be an element of a {{pi}}-system.</ref>
The importance of {{pi}}-systems arises from the fact that if two [[Probability measure|probability measure]]s 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 [[Dynkin system|{{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 <math>\sigma(E_1, E_2, \ldots).</math> So instead we may examine the union of all {{sigma}}-algebras generated by finitely many sets <math display="inline">\bigcup_n \sigma(E_1, \ldots, E_n).</math> This forms a {{pi}}-system that generates the desired {{sigma}}-algebra. Another example is the collection of all [[Interval (mathematics)|intervals]] of the [[Real line|real line]], along with the empty set, which is a {{pi}}-system that generates the very important [[Borel set|Borel {{sigma}}-algebra]] of subsets of the real line.

==Definitions==

A <em>{{pi}}-system</em> is a non-empty collection of sets <math>P</math> that is closed under non-empty finite intersections, which is equivalent to <math>P</math> containing the intersection of any two of its elements. 
If every set in this {{pi}}-system is a subset of <math>\Omega</math> then it is called a {{em|{{pi}}-system on <math>\Omega.</math>}} 

For any non-empty [[Family of sets|family]] <math>\Sigma</math> of subsets of <math>\Omega,</math> there exists a {{pi}}-system <math>\mathcal{I}_{\Sigma},</math> called the '''{{em|{{pi}}-system generated by <math>\boldsymbol\varSigma</math>}}''', that is the unique smallest {{pi}}-system of <math>\Omega</math> containing every element of <math>\Sigma.</math> 
It is equal to the intersection of all {{pi}}-systems containing <math>\Sigma,</math> and can be explicitly described as the set of all possible non-empty finite intersections of elements of <math>\Sigma:</math>
<math display="block">\left\{E_1 \cap \cdots \cap E_n ~:~ 1 \leq n \in \N \text{ and } E_1, \ldots, E_n \in \Sigma\right\}.</math>

A non-empty family of sets has the [[Finite intersection property|finite intersection property]] if and only if the {{pi}}-system it generates does not contain the empty set as an element.

==Examples==

* For any real numbers <math>a</math> and <math>b,</math> the intervals <math>(-\infty, a]</math> form a {{pi}}-system, and the intervals <math>(a, b]</math> form a {{pi}}-system if the empty set is also included. 
* The topology (collection of open subsets) of any [[Topological space|topological space]] is a {{pi}}-system. 
* Every [[Filter (set theory)|filter]] is a {{pi}}-system. Every {{pi}}-system that doesn't contain the empty set is a prefilter (also known as a filter base). 
* For any [[Measurable function|measurable function]] <math>f : \Omega \to \Reals,</math> the set &nbsp;<math>\mathcal{I}_f = \left\{f^{-1}((-\infty, x]) : x \in \Reals\right\}</math> defines a {{pi}}-system, and is called the {{pi}}-system {{em|generated}} by <math>f.</math> (Alternatively, <math>\left\{f^{-1}((a, b]) : a, b \in \Reals, a < b\right\} \cup \{\varnothing\}</math> defines a {{pi}}-system generated by <math>f.</math>)
* If <math>P_1</math> and <math>P_2</math> are {{pi}}-systems for <math>\Omega_1</math> and <math>\Omega_2,</math> respectively, then <math>\{A_1 \times A_2 : A_1 \in P_1, A_2 \in P_2\}</math> is a {{pi}}-system for the [[Cartesian product]] <math>\Omega_1 \times \Omega_2.</math>
* Every {{sigma}}-algebra is a {{pi}}-system.

==Relationship to {{lambda}}-systems==

A [[Dynkin system|{{lambda}}-system]] on <math>\Omega</math> is a set <math>D</math> of subsets of <math>\Omega,</math> satisfying
* <math>\Omega \in D,</math>
* if <math>A \in D</math> then <math>\Omega \setminus A \in D,</math>
* if <math>A_1, A_2, A_3, \ldots</math> is a sequence of [[Disjoint sets|(pairwise) {{em|disjoint}}]] subsets in <math>D</math> then <math>\textstyle\bigcup\limits_{n=1}^\infty A_n \in D.</math>

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

{{anchor|π-λ theorem|Pi-lambda theorem}}
===The {{pi}}-{{lambda}} theorem===

Let <math>D</math> be a {{lambda}}-system, and let &nbsp;<math>\mathcal{I} \subseteq D</math> be a {{pi}}-system contained in <math>D.</math> The {{pi}}-{{lambda}} theorem<ref name="Kallenberg">Kallenberg, Foundations Of Modern Probability, p.&nbsp;2</ref> states that the {{sigma}}-algebra <math>\sigma(\mathcal{I})</math> generated by <math>\mathcal{I}</math> is contained in <math>D ~:~</math> <math>\sigma(\mathcal{I}) \subseteq D.</math>

The {{pi}}-{{lambda}} 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 {{sigma}}-finite measures.<ref name="Durrett">Durrett, Probability Theory and Examples, p.&nbsp;404</ref>

The {{pi}}-{{lambda}} theorem is closely related to the [[Monotone class theorem|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 {{pi}}-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 {{lambda}}-system is often relatively easy. Despite the difference between the two theorems, the {{pi}}-{{lambda}} theorem is sometimes referred to as the monotone class theorem.<ref name="Kallenberg"/>

====Example====

Let <math>\mu_1, \mu_2 : F \to \Reals</math> be two measures on the {{sigma}}-algebra <math>F,</math> and suppose that <math>F = \sigma(I)</math> is generated by a {{pi}}-system <math>I.</math> If
#<math>\mu_1(A) = \mu_2(A)</math> for all <math>A \in I,</math> and 
#<math>\mu_1(\Omega) = \mu_2(\Omega) < \infty,</math>
then <math>\mu_1 = \mu_2.</math>
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 {{sigma}}-algebra, and so the problem of equating measures would be completely hopeless without such a tool.

'''Idea of the proof'''<ref name="Durrett"/>
Define the collection of sets
<math display="block">D = \left\{ A \in \sigma(I) \colon \mu_1(A) = \mu_2(A) \right\}.</math>
By the first assumption, <math>\mu_1</math> and <math>\mu_2</math> agree on <math>I</math> and thus <math>I \subseteq D.</math> By the second assumption, <math>\Omega \in D,</math> and it can further be shown that <math>D</math> is a {{lambda}}-system. It follows from the {{pi}}-{{lambda}} theorem that <math>\sigma(I) \subseteq D \subseteq \sigma(I),</math> and so <math>D = \sigma(I).</math> That is to say, the measures agree on <math>\sigma(I).</math>

=={{pi}}-Systems in probability==

{{pi}}-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 (probability theory)|independence]], though it may also be a consequence of the fact that the {{pi}}-{{lambda}} theorem was proven by the probabilist [[Biography:Eugene Dynkin|Eugene Dynkin]]. Standard measure theory texts typically prove the same results via monotone classes, rather than {{pi}}-systems.

===Equality in distribution===

<onlyinclude>
The {{pi}}-{{lambda}} theorem motivates the common definition of the [[Probability distribution|probability distribution]] of a [[Random variable|random variable]] <math>X : (\Omega, \mathcal F, \operatorname P) \to \Reals</math> in terms of its [[Cumulative distribution function|cumulative distribution function]]. Recall that the cumulative distribution of a random variable  is defined as
<math display="block">F_X(a) = \operatorname{P}[X \leq a], \qquad a \in \Reals,</math>
whereas the seemingly more general {{em|law}} of the variable is the probability measure 
<math display="block">\mathcal{L}_X(B) = \operatorname{P}\left[X^{-1}(B)\right] \quad \text{ for all } B \in \mathcal{B}(\Reals),</math>
where <math>\mathcal{B}(\Reals)</math> is the Borel {{sigma}}-algebra. The random variables <math>X :(\Omega, \mathcal F, \operatorname P) \to \Reals</math> and <math>Y : (\tilde\Omega,\tilde{\mathcal F}, \tilde{\operatorname P}) \to \Reals</math> (on two possibly different [[Probability space|probability space]]s) are {{em|equal in distribution}} (or {{em|law}}), denoted by <math>X \,\stackrel{\mathcal D}{=}\, Y,</math> if they have the same cumulative distribution functions; that is, if <math>F_X = F_Y.</math> The motivation for the definition stems from the observation that if <math>F_X = F_Y,</math> then that is exactly to say that <math>\mathcal{L}_X</math> and <math>\mathcal{L}_Y</math> agree on the {{pi}}-system <math>\{(-\infty, a] : a \in \Reals\}</math> which generates <math>\mathcal{B}(\Reals),</math> and so by the example above: <math>\mathcal{L}_X = \mathcal{L}_Y.</math>

A similar result holds for the joint distribution of a random vector. For example, suppose <math>X</math> and <math>Y</math> are two random variables defined on the same probability space <math>(\Omega, \mathcal{F}, \operatorname{P}),</math> with respectively generated {{pi}}-systems <math>\mathcal{I}_X</math> and <math>\mathcal{I}_Y.</math> The joint cumulative distribution function of <math>(X, Y)</math> is
<math display="block">F_{X,Y}(a, b) = \operatorname{P}[X \leq a, Y \leq b] 
= \operatorname{P}\left[X^{-1}((-\infty, a]) \cap Y^{-1}((-\infty, b])\right], \quad \text{ for all } a, b \in \Reals.</math>

However, <math>A = X^{-1}((-\infty, a]) \in \mathcal{I}_X</math> and <math>B = Y^{-1}((-\infty, b]) \in \mathcal{I}_Y.</math> Because 
<math display="block">\mathcal{I}_{X,Y} = \left\{A \cap B : A \in \mathcal{I}_X, \text{ and } B \in \mathcal{I}_Y\right\}</math>
is a {{pi}}-system generated by the random pair <math>(X, Y),</math> the {{pi}}-{{lambda}} theorem is used to show that the joint cumulative distribution function suffices to determine the joint law of <math>(X, Y).</math> In other words, <math>(X, Y)</math> and <math>(W, Z)</math> have the same distribution if and only if they have the same joint cumulative distribution function.

In the theory of [[Stochastic process|stochastic process]]es, two processes <math>(X_t)_{t \in T}, (Y_t)_{t \in T}</math> are known to be equal in distribution if and only if they agree on all finite-dimensional distributions; that is, for all <math>t_1, \ldots, t_n \in T, \, n \in \N,</math>
<math display="block">\left(X_{t_1}, \ldots, X_{t_n}\right) \,\stackrel{\mathcal{D}}{=}\, \left(Y_{t_1}, \ldots, Y_{t_n}\right).</math>

The proof of this is another application of the {{pi}}-{{lambda}} theorem.<ref>Kallenberg, ''Foundations Of Modern probability'', p.&nbsp;48</ref></onlyinclude>

===Independent random variables===

The theory of {{pi}}-system plays an important role in the probabilistic notion of [[Independence (probability theory)|independence]]. If <math>X</math> and <math>Y</math> are two random variables defined on the same probability space <math>(\Omega, \mathcal{F}, \operatorname{P})</math> then the random variables are independent if and only if their {{pi}}-systems <math>\mathcal{I}_X, \mathcal{I}_Y</math> satisfy for all <math>A \in \mathcal{I}_X</math> and <math>B \in \mathcal{I}_Y,</math>
<math display="block">\operatorname{P}[A \cap B] ~=~ \operatorname{P}[A] \operatorname{P}[B],</math>
which is to say that <math>\mathcal{I}_X, \mathcal{I}_Y</math> are independent. This actually is a special case of the use of {{pi}}-systems for determining the distribution of <math>(X, Y).</math>

====Example====

Let <math>Z = \left(Z_1, Z_2\right),</math> where <math>Z_1, Z_2 \sim \mathcal{N}(0, 1)</math> are [[Independent and identically distributed random variables|iid]] standard normal random variables. Define the radius and argument (arctan) variables
<math display="block">R = \sqrt{Z_1^2 + Z_2^2}, \qquad \Theta = \tan^{-1}\left(Z_2 / Z_1\right).</math>

Then <math>R</math> and <math>\Theta</math> are independent random variables.

To prove this, it is sufficient to show that the {{pi}}-systems <math>\mathcal{I}_R, \mathcal{I}_\Theta</math> are independent: that is, for all <math>\rho \in [0, \infty)</math> and <math>\theta \in [0, 2 \pi],</math>
<math display="block">\operatorname{P}[R \leq \rho, \Theta \leq \theta] = \operatorname{P}[R \leq \rho] \operatorname{P}[\Theta \leq \theta].</math>

Confirming that this is the case is an exercise in changing variables. Fix <math>\rho \in [0, \infty)</math> and <math>\theta \in [0, 2 \pi],</math> then the probability can be expressed as an integral of the probability density function of <math>Z.</math>
<math display="block">\begin{align}
\operatorname P [R \leq \rho, \Theta \leq \theta] &= \int_{R \leq \rho, \, \Theta \leq \theta} \frac{1}{2\pi}\exp\left({-\frac12(z_1^2 + z_2^2)}\right) dz_1 \, dz_2 \\[5pt]
& = \int_0^{\theta} \int_0^\rho \frac{1}{2 \pi}e^{-\frac{r^2}{2}} \; r \, dr \, d\tilde\theta \\[5pt]
& = \left(\int_0^\theta \frac{1}{2 \pi} \, d\tilde \theta\right) \; \left(\int_0^\rho e^{-\frac{r^2}{2}} \; r \, dr\right) \\[5pt]
& = \operatorname P[\Theta \leq \theta]\operatorname P[R \leq \rho].
\end{align}</math>

==See also==


* {{annotated link|Delta-ring|{{delta}}-ring}}
* {{annotated link|Field of sets}}
* {{annotated link|Ideal (set theory)}}
* {{annotated link|Independence (probability theory)}}
* {{annotated link|Dynkin system|{{lambda}}-system (Dynkin system)}}
* {{annotated link|Monotone class theorem}}
* {{annotated link|Probability distribution}}
* {{annotated link|Ring of sets}}
* {{annotated link|Σ-algebra}}
* {{annotated link|Sigma-ideal|{{sigma}}-ideal}}
* {{annotated link|Sigma-ring|{{sigma}}-ring}}

==Notes==

{{reflist|group=nb}}

==Citations==

{{reflist|50em}}

==References==

* {{cite book|last=Gut|first=Allan|title=Probability: A Graduate Course|year=2005|series=Springer Texts in Statistics|publisher=Springer|location=New York|doi=10.1007/b138932|isbn=0-387-22833-0}}
* {{cite book|last=Williams|first=David|title=Probability with Martingales|year=1991|publisher=Cambridge University Press|isbn=0-521-40605-6}}
* {{Durrett Probability Theory and Examples 5th Edition}} <!--{{sfn|Durrett|2019|p=}}-->


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