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
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.
that is closed under non-empty finite intersections, which is equivalent to
and can be explicitly described as the set of all possible non-empty finite intersections of elements of
However, a useful classification is that any set system which is both a 𝜆-system and a π-system is a 𝜎-algebra.
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.
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
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.
The π-𝜆 theorem motivates the common definition of the probability distribution of a random variable
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
(on two possibly different probability spaces) are equal in distribution (or law), denoted by
A similar result holds for the joint distribution of a random vector.
are two random variables defined on the same probability space
[3] The theory of π-system plays an important role in the probabilistic notion of independence.
are two random variables defined on the same probability space
This actually is a special case of the use of π-systems for determining the distribution of
are iid standard normal random variables.
Define the radius and argument (arctan) variables
Confirming that this is the case is an exercise in changing variables.