Pi-system

In mathematics, a π-system on a set Ω is a set P, consisting of certain subsets of Ω, such that

• P is non-empty.
• A ∩ B ∈ P whenever A and B are in P.

That is, P is a non-empty family of subsets of Ω that is closed under finite intersections.

Properties

• For any subset Σ ⊆ Ω, there exists a π-system $\mathcal I_\left\{\Sigma\right\}$ which is the unique smallest π-system of Ω to contain every element of Σ, and is called the π-system generated by Σ.
• For any measurable function $f \colon \Omega \rightarrow \mathbb\left\{R\right\}$, the set $\mathcal\left\{I\right\}_f = \left \\left\{ f^\left\{-1\right\}\left\left(\left\left( - \infty, x \right\right]\right\right) \colon x \in \mathbb\left\{R\right\} \right \\right\}$ defines a π-system, and is called the π-system generated by f.
• Any σ-algebra is a π-system.

Relationship to λ-Systems

A λ-system on Ω is a set D of subsets of Ω, satisfying

• Ω ∈ $D$,
• if A, B ∈ $D$ and A ⊆ B, then B \ A ∈ $D$,
• if A1, A2, A3, ... is a sequence of subsets in $D$ and An ⊆ An+1 for all n ≥ 1, then $\cup_\left\{n=1\right\}^\left\{\infty\right\}A_n\in D$.

Whilst it is true that any σ-algebra satisfies the properties of being a π-system and a λ-system, in general 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

Let $D$ be a λ-system, and let $\mathcal\left\{I\right\} \subseteq D$ be a π-system contained in $D$. The π-λ Theorem[1] states that the σ-algebra $\sigma\left( \mathcal\left\{I\right\}\right)$ generated by $\mathcal\left\{I\right\}$ is contained in $D$: $\sigma\left( \mathcal\left\{I\right\}\right) \subset D$.

The π-λ theorem can be used to prove many elementary measure theoretic results. For instance, it is used in proving the existence 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. Whilst the two theorems are different, the π-λ theorem is sometimes referred to as the monotone class theorem.[1]

Example

Let μ1 , μ2 : F → R be two measures on the σ-algebra F, and suppose that F = σ(I) is generated by a π-system I. If

1. μ1(A) = μ2(A), A I, and
2. μ1(Ω) < , μ2(Ω) < ,

then μ1 = μ2. This is the uniqueness statement of the Carathéodory extension theorem for finite measures.

Idea of Proof[2] Define the collection of sets

$D = \left\\left\{ A \in \sigma\left(I\right) \colon \mu_1\left(A\right) = \mu_\left(A\right) \right\\right\}.$

It can be shown that D defines a λ-system. By our first assumption μ1 andμ2 agree on I it follows that I D. It follows from the π-λ theorem that σ(I) D σ(I), and so D = σ(I). That is to say the measures agree on σ(I)

Π-Systems in Probability

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 the probabilistic notion of independence, though may also be a consequence of the fact that the π-λ theorem was proven by the probabilist Eugene Dynkin. Standard measure theory texts will prove the same results via monotone classes, rather than π-systems.

Equality in Distribution

The π-λ theorem motivates the definition of equality in distribution. Recalling that the cumulative distribution function of a random variable $X \colon\left(\Omega, \mathcal F, \mathbb P\right) \rightarrow \mathbb R$ is defined as

$F_X\left(a\right) = \mathbb\left\{P\right\}\left\left[ X \leq a \right\right], \qquad a \in \mathbb\left\{R\right\}$,

and the law of the variable is the probability measure

$\mathcal\left\{L\right\}_X\left(B\right) = \mathbb\left\{P\right\}\left\left[ X^\left\{-1\right\}\left(B\right) \right\right], \qquad B \in \mathcal\left\{B\right\}\left(\mathbb R\right)$,

where $\mathcal\left\{B\right\}\left(\mathbb R\right)$ is the Borel σ-algebra. We say that the random variables $X \colon\left(\Omega, \mathcal F, \mathbb P\right)$, and $Y \colon\left(\tilde\Omega,\tilde\left\{ \mathcal F\right\}, \tilde\left\{\mathbb P\right\}\right) \rightarrow \mathbb R$ (on two possibly different probability spaces) are equal in distribution (or law), $X \stackrel\left\{\mathcal D\right\}\left\{=\right\} Y$, if they have the same cumulative distribution functions, FX = FY. The motivation for the definition stems from the observation that if FX = FY, then that is exactly to say that $\mathcal\left\{L\right\}_X$ and $\mathcal\left\{L\right\}_Y$ agree on the π-system $\left\\left\{\left(-\infty, a\right] \colon a \in \mathbb R \right\\right\}$ which generates $\mathcal\left\{B\right\}\left(\mathbb R\right)$, and so by the example above: $\mathcal\left\{L\right\}_X = \mathcal\left\{L\right\}_Y$.

In the theory of stochastic processes, two processes $\left(X_t\right)_\left\{t \in T\right\}, \left(Y_t\right)_\left\{t \in T\right\}$ are known to be equal in distribution if and only if they agree on all finite dimmensional distributions. i.e. for all $t_1,\ldots, t_n \in T, \, n \in \mathbb N$

$\left(X_\left\{t_1\right\},\ldots, X_\left\{t_n\right\}\right) \stackrel\left\{\mathcal\left\{D\right\}\right\}\left\{=\right\} \left(Y_\left\{t_1\right\},\ldots, Y_\left\{t_n\right\}\right)$.

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

Independent Random Variables

The theory of π-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 $\left(\Omega, \mathcal\left\{F\right\}, \mathbb\left\{P\right\}\right)$ then the random variables are independent if and only if their π-systems $\mathcal\left\{I\right\}_X, \mathcal\left\{I\right\}_Y$ satisfy

$\mathbb\left\{P\right\}\left\left[ A \cap B \right\right] = \mathbb\left\{P\right\}\left\left[ A \right\right] \mathbb\left\{P\right\}\left\left[ B \right\right], \qquad \forall A \in \mathcal\left\{I\right\}_X, \, B \in \mathcal\left\{I\right\}_Y,$

which is to say that $\mathcal\left\{I\right\}_X, \mathcal\left\{I\right\}_Y$ are independent.

Example

Let $Z = \left(Z_1, Z_2\right)$, where $Z_1, Z_2 \sim \mathcal\left\{N\right\}\left(0,1\right)$ are iid standard normal random variables. Define the radius and argument variables

$R = \sqrt\left\{Z_1^2 + Z_2^2\right\}, \qquad \Theta = \tan^\left\{-1\right\}\left(Z_2/Z_1\right)$.

Then $R$ and $\Theta$ are independent random variables.

To prove this, it is sufficient to show that the π-systems $\mathcal\left\{I\right\}_R, \mathcal\left\{I\right\}_\Theta$ are independent: i.e.

$\mathbb P \left[ R \leq \rho, \Theta \leq \theta\right] = \mathbb P\left[R \leq \rho\right] \mathbb P\left[\Theta \leq \theta\right] \quad \forall \rho \in \left[0,\infty\right), \, \theta \in \left[0,2\pi\right].$

Confirming that this is the case is an exercise in changing variables. Fix $\rho \in \left[0,\infty\right), \, \theta \in \left[0,2\pi\right]$, then the probability can be expressed as an integral of the probability density function of $Z$

\begin\left\{align\right\} \mathbb P \left[ R \leq \rho, \Theta \leq \theta\right] &= \int_\left\{R \leq \rho, \, \Theta \leq \theta\right\} \frac\left\{1\right\}\left\{2\pi\right\}\exp\left\left(\left\{-\frac12\left(z_1^2 + z_2^2\right)\right\}\right\right) dz_1dz_2 \\

& = \int_0^\theta \int_0^\rho \frac{1}{2\pi}e^{-\frac{r^2}{2}}r dr d\tilde\theta \\ & = \left( \int_0^\theta \frac{1}{2\pi}d\tilde \theta \right) \left( \int_0^\rho e^{-\frac{r^2}{2}}r dr\right) \\ & = \mathbb P[\Theta \leq \theta]\mathbb P[R \leq \rho] \end{align}