In mathematics, a π-system (or pi-system) on a set Ω is a collection P of certain subsets of Ω, such that
Contents
- Examples
- Relationship to Systems
- The Theorem
- Example
- Systems in Probability
- Equality in Distribution
- Independent Random Variables
- References
That is, P is a non-empty family of subsets of Ω that is closed under finite intersections. The importance of π-systems arise 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
Examples
Relationship to λ-Systems
A λ-system on Ω is a set D of subsets of Ω, satisfying
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
Let
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.
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.
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
- μ1(A) = μ2(A), ∀ A ∈ I, and
- μ1(Ω) = μ2(Ω) < ∞,
then μ1 = μ2. 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 Proof Define the collection of sets
By the first assumption, μ1 and μ2 agree on I and thus I ⊆ D. By the second assumption, Ω ∈ D, and it can further be shown that D is a λ-system. 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 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
The π-λ theorem motivates the common definition of the probability distribution of a random variable
whereas the seemingly more general law of the variable is the probability measure
where
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
However,
is a π-system generated by the random pair (X,Y), the π-λ 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
The proof of this is another application of the π-λ theorem.
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
which is to say that
Example
Let
Then
To prove this, it is sufficient to show that the π-systems
Confirming that this is the case is an exercise in changing variables. Fix