Pi-system explained

\Omega is a collection P of certain subsets of \Omega, such thatP 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 \Omega that is closed under non-empty finite intersections.^[1] The importance of -systems arises from the fact that if two probability measures agree on a -system, then they agree on the -algebra generated by that -system. Moreover, if other properties, such as equality of integrals, hold for the -system, then they hold for the generated -algebra as well. This is the case whenever the collection of subsets for which the property holds is a -system. -systems are also useful for checking independence of random variables.

This is desirable because in practice, -systems are often simpler to work with than -algebras. For example, it may be awkward to work with -algebras generated by infinitely many sets

\sigma(E _{1,} E _{2,} \ldots). So instead we may examine the union of all -algebras generated by finitely many sets $\backslash bigcup\_n\; \backslash sigma(E\_1,\; \backslash ldots,\; E\_n).$ This forms a -system that generates the desired -algebra. Another example is the collection of all intervals of the real line, along with the empty set, which is a -system that generates the very important Borel -algebra of subsets of the real line.

Definitions

A -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 -system is a subset of \Omega then it is called a \Sigma of subsets of \Omega, there exists a -system l{I} _{\Sigma} , called the , that is the unique smallest -system of \Omega containing every element of \Sigma. It is equal to the intersection of all -systems containing \Sigma, and can be explicitly described as the set of all possible non-empty finite intersections of elements of \Sigma: $$\backslash left\backslash .$$

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

Examples

• For any real numbers

a and b, the intervals ( -inf ty, a] form a -system, and the intervals (a, b] form a -system if the empty set is also included.

• The topology (collection of open subsets) of any topological space is a -system.
• Every filter is a -system. Every -system that doesn't contain the empty set is a prefilter (also known as a filter base).

f : \Omega \to \Reals, the set  l{I} _{f} = \left\{f ^{-1} (( -inf ty, x]) : x \in \Reals\right\} defines a -system, and is called the -system by f. (Alternatively, \left\{f ^{-1} ((a, b]) : a, b \in \Reals, a < b\right\} \cup \{\varnothing\} defines a -system generated by f. )

• If

P _{1} and P _{2} are -systems for \Omega _{1} and \Omega _{2,} respectively, then \{A _{1} x A _{2} : A _{1} \in P _{1,} A _{2} \in P _{2\}} is a -system for the Cartesian product \Omega _{1} x \Omega _{2.}

• Every -algebra is a -system.

Relationship to -systems

A -system on

\Omega is a set D of subsets of \Omega, satisfying\Omega \in D,

• if

A \in D then \Omega \setminus A \in D,

• if

A _{1,} A _{2,} A _{3,} \ldots is a sequence of (pairwise) subsets in D then style c up\limits ^{inf ty} _{n =1} A _{n} \in D.

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

The - theorem

Let

D be a -system, and let  l{I} \subseteq D be a -system contained in D. The - theorem^[2] states that the -algebra \sigma( l{I}) generated by l{I} is contained in D ~:~ \sigma( l{I}) \subseteq D.

The - 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 -finite measures.^[3]

The - 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 -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 -system is often relatively easy. Despite the difference between the two theorems, the - theorem is sometimes referred to as the monotone class theorem.^[2]

Example

Let

\mu _{1,} \mu _{2} : F \to \Reals be two measures on the -algebra F, and suppose that F = \sigma(I) is generated by a -system I. If\mu _{1(A)} = \mu _{2(A)} for all A \in I, and \mu _{1(\Omega)} = \mu _{2(\Omega)} < inf ty, then \mu _{1} = \mu _{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 -algebra, and so the problem of equating measures would be completely hopeless without such a tool.

Idea of the proof^[3] Define the collection of sets$$D\; =\; \backslash left\backslash .$$By the first assumption,

\mu _{1} and \mu _{2} agree on I and thus I \subseteq D. By the second assumption, \Omega \in D, and it can further be shown that D is a -system. It follows from the - theorem that \sigma(I) \subseteq D \subseteq \sigma(I), and so D = \sigma(I). That is to say, the measures agree on \sigma(I).

-Systems in probability

-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 - theorem was proven by the probabilist Eugene Dynkin. Standard measure theory texts typically prove the same results via monotone classes, rather than -systems.

Equality in distribution

X : (\Omega, l F, \operatorname P) \to \Reals in terms of its cumulative distribution function. Recall that the cumulative distribution of a random variable is defined as$$F\_X(a)\; =\; \backslash operatorname[X\; \backslash leq\; a],\; \backslash qquad\; a\; \backslash in\; \backslash Reals,$$whereas the seemingly more general of the variable is the probability measure $$\backslash mathcal\_X(B)\; =\; \backslash operatorname\backslash left[X^\{-1\}(B)\backslash right]\; \backslash quad\; \backslash text\; B\; \backslash in\; \backslash mathcal(\backslash Reals),$$where l{B}(\Reals) is the Borel -algebra. The random variables X :(\Omega, l F, \operatorname P) \to \Reals and Y : (\tilde\Omega,\tilde{ l F}, \tilde{\operatorname P}) \to \Reals (on two possibly different probability spaces) are (or), denoted by X \stackrel{ l 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{L} _{X} and l{L} _{Y} agree on the -system \{( -inf ty, a] : a \in \Reals\} which generates l{B}(\Reals), and so by the example above: l{L} _{X} = l{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 (\Omega, l{F}, \operatorname{P}), with respectively generated -systems l{I} _{X} and l{I} _{Y.} The joint cumulative distribution function of (X, Y) is$$F\_(a,\; b)\; =\; \backslash operatorname[X\; \backslash leq\; a,\; Y\; \backslash leq\; b]\; =\; \backslash operatorname\backslash left[X^\{-1\}((-\backslash infty,\; a])\; \backslash cap\; Y^((-\backslash infty,\; b])\backslash right],\; \backslash quad\; \backslash text\; a,\; b\; \backslash in\; \backslash Reals.$$

However,

A = X ^{-1} (( -inf ty, a]) \in l{I} _{X} and B = Y ^{-1} (( -inf ty, b]) \in l{I} _{Y.} Because $$\backslash mathcal\_\; =\; \backslash left\backslash$$is a -system generated by the random pair (X, Y), the - 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} , (Y _{t)} _{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,} \ldots, t _{n} \in T, n \in \N, $$\backslash left(X\_,\; \backslash ldots,\; X\_\backslash right)\; \backslash ,\backslash stackrel\backslash ,\; \backslash left(Y\_,\; \backslash ldots,\; Y\_\backslash right).$$

The proof of this is another application of the - theorem.^[4]

Independent random variables

The theory of -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 (\Omega, l{F}, \operatorname{P}) then the random variables are independent if and only if their -systems l{I} _{X,} l{I} _{Y} satisfy for all A \in l{I} _{X} and B \in l{I} _{Y,} $$\backslash operatorname[A\; \backslash cap\; B]\; ~=~\; \backslash operatorname[A]\; \backslash operatorname[B],$$which is to say that l{I} _{X,} l{I} _{Y} are independent. This actually is a special case of the use of -systems for determining the distribution of (X, Y).

Example

Let

Z = \left(Z _{1,} Z _{2\right),} where Z _{1,} Z _{2} \sim l{N}(0, 1 ) are iid standard normal random variables. Define the radius and argument (arctan) variables$$R\; =\; \backslash sqrt,\; \backslash qquad\; \backslash Theta\; =\; \backslash tan^\backslash left(Z\_2\; /\; Z\_1\backslash right).$$

Then

R and \Theta are independent random variables.

To prove this, it is sufficient to show that the -systems

l{I} _{R,} l{I} _{\Theta} are independent: that is, for all \rho \in [ 0 , inf ty) and \theta \in [ 0 , 2 \pi], $$\backslash operatorname[R\; \backslash leq\; \backslash rho,\; \backslash Theta\; \backslash leq\; \backslash theta]\; =\; \backslash operatorname[R\; \backslash leq\; \backslash rho]\; \backslash operatorname[\backslash Theta\; \backslash leq\; \backslash theta].$$

Confirming that this is the case is an exercise in changing variables. Fix

\rho \in [ 0 , inf ty) and \theta \in [ 0 , 2 \pi], then the probability can be expressed as an integral of the probability density function of Z.

References

• Book: Gut, Allan. Allan Gut. Probability: A Graduate Course. 2005. Springer Texts in Statistics. Springer. New York. 10.1007/b138932. 0-387-22833-0.
• Book: Williams, David. David Williams (mathematician). Probability with Martingales. 1991. Cambridge University Press. 0-521-40605-6.

Notes and References

1. The nullary (0-ary) intersection of subsets of \Omega is by convention equal to \Omega, which is not required to be an element of a -system.
2. Kallenberg, Foundations Of Modern Probability, p. 2
3. Durrett, Probability Theory and Examples, p. 404
4. Kallenberg, Foundations Of Modern Probability, p. 48