Pi-system

Template:Short description Template:This In mathematics, a Template:Pi-system (or pi-system) on a set $Ω$ is a collection $P$ of certain subsets of $Ω,$ such that

$P$ is non-empty.
If $A, B \in P$ then $A \cap B \in P .$

That is, $P$ is a non-empty family of subsets of $Ω$ that is closed under non-empty finite intersections.^{[nb 1]} The importance of Template:Pi-systems arises from the fact that if two probability measures agree on a Template:Pi-system, then they agree on the [[Sigma algebra|Template:Sigma-algebra]] generated by that Template:Pi-system. Moreover, if other properties, such as equality of integrals, hold for the Template:Pi-system, then they hold for the generated Template:Sigma-algebra as well. This is the case whenever the collection of subsets for which the property holds is a [[Dynkin system|Template:Lambda-system]]. Template:Pi-systems are also useful for checking independence of random variables.

This is desirable because in practice, Template:Pi-systems are often simpler to work with than Template:Sigma-algebras. For example, it may be awkward to work with Template:Sigma-algebras generated by infinitely many sets $σ (E_{1}, E_{2}, \dots) .$ So instead we may examine the union of all Template:Sigma-algebras generated by finitely many sets $⋃_{n} σ (E_{1}, \dots, E_{n}) .$ This forms a Template:Pi-system that generates the desired Template:Sigma-algebra. Another example is the collection of all intervals of the real line, along with the empty set, which is a Template:Pi-system that generates the very important [[Borel set|Borel Template:Sigma-algebra]] of subsets of the real line.

Definitions

A Template:Pi-system is a non-empty collection of sets $P$ that is closed under non-empty finite intersections, which is equivalent to $P$ containing the intersection of any two of its elements. If every set in this Template:Pi-system is a subset of $Ω$ then it is called a Template:Em

For any non-empty family $Σ$ of subsets of $Ω,$ there exists a Template:Pi-system $ℐ_{Σ},$ called the Template:Em, that is the unique smallest Template:Pi-system of $Ω$ containing every element of $Σ .$ It is equal to the intersection of all Template:Pi-systems containing $Σ,$ and can be explicitly described as the set of all possible non-empty finite intersections of elements of $Σ :$ ${E_{1} \cap \dots \cap E_{n} : 1 \leq n \in ℕ and E_{1}, \dots, E_{n} \in Σ} .$

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

Examples

For any real numbers $a$ and $b,$ the intervals $(- \infty, a]$ form a Template:Pi-system, and the intervals $(a, b]$ form a Template:Pi-system if the empty set is also included.
The topology (collection of open subsets) of any topological space is a Template:Pi-system.
Every filter is a Template:Pi-system. Every Template:Pi-system that doesn't contain the empty set is a prefilter (also known as a filter base).
For any measurable function $f : Ω \to ℝ,$ the set $ℐ_{f} = {f^{- 1} ((- \infty, x]) : x \in ℝ}$ defines a Template:Pi-system, and is called the Template:Pi-system Template:Em by $f .$ (Alternatively, ${f^{- 1} ((a, b]) : a, b \in ℝ, a < b} \cup {\emptyset}$ defines a Template:Pi-system generated by $f .$ )
If $P_{1}$ and $P_{2}$ are Template:Pi-systems for $Ω_{1}$ and $Ω_{2},$ respectively, then ${A_{1} \times A_{2} : A_{1} \in P_{1}, A_{2} \in P_{2}}$ is a Template:Pi-system for the Cartesian product $Ω_{1} \times Ω_{2} .$
Every Template:Sigma-algebra is a Template:Pi-system.

Relationship to Template:Lambda-systems

A [[Dynkin system|Template:Lambda-system]] on $Ω$ is a set $D$ of subsets of $Ω,$ satisfying

$Ω \in D,$
if $A \in D$ then $Ω ∖ A \in D,$
if $A_{1}, A_{2}, A_{3}, \dots$ is a sequence of [[Disjoint sets|(pairwise) Template:Em]] subsets in $D$ then $⋃_{n = 1}^{\infty} A_{n} \in D .$

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

Template:Anchor

The Template:Pi-Template:Lambda theorem

Template:See also

Let $D$ be a Template:Lambda-system, and let $ℐ \subseteq D$ be a Template:Pi-system contained in $D .$ The Template:Pi-Template:Lambda theorem^[1] states that the Template:Sigma-algebra $σ (ℐ)$ generated by $ℐ$ is contained in $D :$ $σ (ℐ) \subseteq D .$

The Template:Pi-Template: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 Template:Sigma-finite measures.^[2]

The Template:Pi-Template:Lambda 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 Template: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 Template:Lambda-system is often relatively easy. Despite the difference between the two theorems, the Template:Pi-Template:Lambda theorem is sometimes referred to as the monotone class theorem.^[1]

Example

Let $μ_{1}, μ_{2} : F \to ℝ$ be two measures on the Template:Sigma-algebra $F,$ and suppose that $F = σ (I)$ is generated by a Template:Pi-system $I .$ If

$μ_{1} (A) = μ_{2} (A)$ for all $A \in I,$ and
$μ_{1} (Ω) = μ_{2} (Ω) < \infty,$

then $μ_{1} = μ_{2} .$ 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 Template:Sigma-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 $D = {A \in σ (I) : μ_{1} (A) = μ_{2} (A)} .$ By the first assumption, $μ_{1}$ and $μ_{2}$ agree on $I$ and thus $I \subseteq D .$ By the second assumption, $Ω \in D,$ and it can further be shown that $D$ is a Template:Lambda-system. It follows from the Template:Pi-Template:Lambda theorem that $σ (I) \subseteq D \subseteq σ (I),$ and so $D = σ (I) .$ That is to say, the measures agree on $σ (I) .$

Template:Pi-Systems in probability

Template: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, though it may also be a consequence of the fact that the Template:Pi-Template:Lambda theorem was proven by the probabilist Eugene Dynkin. Standard measure theory texts typically prove the same results via monotone classes, rather than Template:Pi-systems.

Equality in distribution

The Template:Pi-Template:Lambda theorem motivates the common definition of the probability distribution of a random variable $X : (Ω, ℱ, P) \to ℝ$ in terms of its cumulative distribution function. Recall that the cumulative distribution of a random variable is defined as $F_{X} (a) = P [X \leq a], a \in ℝ,$ whereas the seemingly more general Template:Em of the variable is the probability measure $ℒ_{X} (B) = P [X^{- 1} (B)] for all B \in ℬ (ℝ),$ where $ℬ (ℝ)$ is the Borel Template:Sigma-algebra. The random variables $X : (Ω, ℱ, P) \to ℝ$ and $Y : (\tilde{Ω}, \tilde{ℱ}, \tilde{P}) \to ℝ$ (on two possibly different probability spaces) are Template:Em (or Template:Em), denoted by $X \overset{𝒟}{=} Y,$ if they have the same cumulative distribution functions; that is, if $F_{X} = F_{Y} .$ The motivation for the definition stems from the observation that if $F_{X} = F_{Y},$ then that is exactly to say that $ℒ_{X}$ and $ℒ_{Y}$ agree on the Template:Pi-system ${(- \infty, a] : a \in ℝ}$ which generates $ℬ (ℝ),$ and so by the example above: $ℒ_{X} = ℒ_{Y} .$

A similar result holds for the joint distribution of a random vector. For example, suppose $X$ and $Y$ are two random variables defined on the same probability space $(Ω, ℱ, P),$ with respectively generated Template:Pi-systems $ℐ_{X}$ and $ℐ_{Y} .$ The joint cumulative distribution function of $(X, Y)$ is $F_{X, Y} (a, b) = P [X \leq a, Y \leq b] = P [X^{- 1} ((- \infty, a]) \cap Y^{- 1} ((- \infty, b])], for all a, b \in ℝ .$

However, $A = X^{- 1} ((- \infty, a]) \in ℐ_{X}$ and $B = Y^{- 1} ((- \infty, b]) \in ℐ_{Y} .$ Because $ℐ_{X, Y} = {A \cap B : A \in ℐ_{X}, and B \in ℐ_{Y}}$ is a Template:Pi-system generated by the random pair $(X, Y),$ the Template:Pi-Template:Lambda theorem is used to show that the joint cumulative distribution function suffices to determine the joint law of $(X, Y) .$ In other words, $(X, Y)$ and $(W, Z)$ have the same distribution if and only if they have the same joint cumulative distribution function.

In the theory of stochastic processes, two processes $(X_{t})_{t \in T}, (Y_{t})_{t \in T}$ are known to be equal in distribution if and only if they agree on all finite-dimensional distributions; that is, for all $t_{1}, \dots, t_{n} \in T, n \in ℕ,$ $(X_{t_{1}}, \dots, X_{t_{n}}) \overset{𝒟}{=} (Y_{t_{1}}, \dots, Y_{t_{n}}) .$

The proof of this is another application of the Template:Pi-Template:Lambda theorem.^[3]

Independent random variables

The theory of Template:Pi-system plays an important role in the probabilistic notion of independence. If $X$ and $Y$ are two random variables defined on the same probability space $(Ω, ℱ, P)$ then the random variables are independent if and only if their Template:Pi-systems $ℐ_{X}, ℐ_{Y}$ satisfy for all $A \in ℐ_{X}$ and $B \in ℐ_{Y},$ $P [A \cap B] = P [A] P [B],$ which is to say that $ℐ_{X}, ℐ_{Y}$ are independent. This actually is a special case of the use of Template:Pi-systems for determining the distribution of $(X, Y) .$

Example

Let $Z = (Z_{1}, Z_{2}),$ where $Z_{1}, Z_{2} \sim 𝒩 (0, 1)$ are iid standard normal random variables. Define the radius and argument (arctan) variables $R = \sqrt{Z_{1}^{2} + Z_{2}^{2}}, Θ = \tan^{- 1} (Z_{2} / Z_{1}) .$

Then $R$ and $Θ$ are independent random variables.

To prove this, it is sufficient to show that the Template:Pi-systems $ℐ_{R}, ℐ_{Θ}$ are independent: that is, for all $ρ \in [0, \infty)$ and $θ \in [0, 2 π],$ $P [R \leq ρ, Θ \leq θ] = P [R \leq ρ] P [Θ \leq θ] .$

Confirming that this is the case is an exercise in changing variables. Fix $ρ \in [0, \infty)$ and $θ \in [0, 2 π],$ then the probability can be expressed as an integral of the probability density function of $Z .$ $\begin{matrix} P [R \leq ρ, Θ \leq θ] & = \int_{R \leq ρ, Θ \leq θ} \frac{1}{2 π} \exp (- \frac{1}{2} (z_{1}^{2} + z_{2}^{2})) d z_{1} d z_{2} \\ = \int_{0}^{θ} \int_{0}^{ρ} \frac{1}{2 π} e^{- \frac{r^{2}}{2}} r d r d \tilde{θ} \\ = (\int_{0}^{θ} \frac{1}{2 π} d \tilde{θ}) (\int_{0}^{ρ} e^{- \frac{r^{2}}{2}} r d r) \\ = P [Θ \leq θ] P [R \leq ρ] . \end{matrix}$

Template:Large Template:Navbar
Template:Nowrap Template:Nowrap	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small
Template:Nowrap	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na
Template:Nowrap	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Never
Template:Nowrap	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Never
[[Monotone class\|Template:Nowrap]]	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na
[[Dynkin system\|Template:Nowrap Template:Nowrap]]	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Ya	Template:Ya	Template:Ya	Never
[[Ring of sets\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na
[[Ring of sets\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Never
[[Delta-ring\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Ya	Never
[[Sigma-ring\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Ya	Never
[[Field of sets\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Ya	Never
[[σ-algebra\|Template:Nowrap Template:Small]]	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Never
[[Dual ideal\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Na
[[Filter (set theory)\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Never	Never	Template:Na	Template:Ya	Template:Ya	$\emptyset \in̸ ℱ$	Template:Ya
[[Prefilter\|Template:Nowrap Template:Small]]	Template:Ya	Template:Na	Template:Na	Never	Never	Template:Na	Template:Na	Template:Na	$\emptyset \in̸ ℱ$	Template:Ya
[[Filter subbase\|Template:Nowrap]]	Template:Na	Template:Na	Template:Na	Never	Never	Template:Na	Template:Na	Template:Na	$\emptyset \in̸ ℱ$	Template:Ya
[[Topology (structure)\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Ya	Never
[[Topology (structure)\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Na	Template:Ya	Template:Ya	Never
Template:Nowrap Template:Nowrap	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small
Additionally, a Template:Em is a [[pi-system\|Template:Pi-system]] where every complement $B ∖ A$ is equal to a finite disjoint union of sets in $ℱ .$ A Template:Em is a semiring where every complement $Ω ∖ A$ is equal to a finite disjoint union of sets in $ℱ .$ $A, B, A_{1}, A_{2}, \dots$ are arbitrary elements of $ℱ$ and it is assumed that $ℱ \neq \emptyset .$

Notes

Template:Reflist

Citations

Template:Reflist

References

Template:Measure theory

Cite error: <ref> tags exist for a group named "nb", but no corresponding <references group="nb"/> tag was found

↑ ^1.0 ^1.1 Kallenberg, Foundations Of Modern Probability, p. 2
↑ ^2.0 ^2.1 Durrett, Probability Theory and Examples, p. 404
↑ Kallenberg, Foundations Of Modern Probability, p. 48

[Kallenberg-2] 1.0 ^1.1 Kallenberg, Foundations Of Modern Probability, p. 2

[Durrett-3] 2.0 ^2.1 Durrett, Probability Theory and Examples, p. 404

[4] Kallenberg, Foundations Of Modern Probability, p. 48

[nb 1]

[1]

[2]

[3]

Pi-system

Contents

Definitions

Examples

Relationship to Template:Lambda-systems

The Template:Pi-Template:Lambda theorem

Example

Template:Pi-Systems in probability

Equality in distribution

Independent random variables

Example

See also

Notes

Citations

References

Navigation menu

Pi-system

Definitions

Examples

Relationship to Template:Lambda-systems

The Template:Pi-Template:Lambda theorem

Example

Template:Pi-Systems in probability

Equality in distribution

Independent random variables

Example

See also

Notes

Citations

References

Navigation menu

Search