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In mathematics, a -system (or pi-system) on a set \Omega is a collection P of certain
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s of \Omega, such that * P is
non-empty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
. * 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 Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
under non-empty finite intersections.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. The importance of -systems arises from the fact that if two
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...
s 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 \bigcup_n \sigma(E_1, \ldots, E_n). This forms a -system that generates the desired -algebra. Another example is the collection of all
intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets * A statistical level of measurement * Interval est ...
of the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
, 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 For any non-empty
family Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Ideal ...
\Sigma of subsets of \Omega, there exists a -system \mathcal_, 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: \left\. A non-empty family of sets has the
finite intersection property In general topology, a branch of mathematics, a non-empty family ''A'' of subsets of a set X is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of A is non-empty. It has the strong finite inter ...
if and only if the -system it generates does not contain the empty set as an element.


Examples

* For a, b \in \R, the intervals (-\infty, a] form a -system, and the intervals (a, b] form a -system if the empty set is also included. * The Topological space, topology (collection of
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s) of any
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
is a -system. * Every
filter Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component that ...
is a -system. Every -system that doesn't contain the empty set is a
prefilter In mathematics, a filter on a set X is a family \mathcal of subsets such that: # X \in \mathcal and \emptyset \notin \mathcal # if A\in \mathcal and B \in \mathcal, then A\cap B\in \mathcal # If A,B\subset X,A\in \mathcal, and A\subset B, then ...
(also known as a filter base). * For any
measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in d ...
f : \Omega \to \R, the set  \mathcal_f = \left\ defines a -system, and is called the -system by f. (Alternatively, \left\ \cup \ defines a -system generated by f.) * If P_1 and P_2 are -systems for \Omega_1 and \Omega_2, respectively, then \ is a -system for the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
\Omega_1 \times \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 A^c \in D (where A^c := \Omega \setminus A), * if A_1, A_2, A_3, \dots is a sequence of (pairwise) subsets in D then \cup_^ 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  \mathcal \subseteq D be a -system contained in D. The - theoremKallenberg, Foundations Of Modern Probability, p. 2 states that the -algebra \sigma( \mathcal) generated by \mathcal is contained in D ~:~ \sigma(\mathcal) \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.Durrett, Probability Theory and Examples, p. 404 The - theorem is closely related to the
monotone class theorem In measure theory and probability, the monotone class theorem connects monotone classes and sigma-algebras. The theorem says that the smallest monotone class containing an algebra of sets G is precisely the smallest -algebra containing G. It ...
, 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.


Example

Let \mu_1, \mu_2 : F \to \R 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) < \infty, 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 Define the collection of sets D = \left\. 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 Independence is a condition of a person, nation, country, or state in which residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the st ...
, though it may also be a consequence of the fact that the - theorem was proven by the probabilist
Eugene Dynkin Eugene Borisovich Dynkin (russian: link=no, Евгений Борисович Дынкин; 11 May 1924 – 14 November 2014) was a Soviet and American mathematician. He made contributions to the fields of probability and algebra, especially sem ...
. Standard measure theory texts typically prove the same results via monotone classes, rather than -systems.


Equality in distribution

The - theorem motivates the common definition of the
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
X : (\Omega, \mathcal F, \operatorname P) \to \Reals in terms of its
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
. Recall that the cumulative distribution of a random variable is defined as F_X(a) = \operatorname X \leq a \qquad a \in \R, whereas the seemingly more general of the variable is the probability measure \mathcal_X(B) = \operatorname\left X^(B) \right\quad \text B \in \mathcal(\Reals), where \mathcal(\Reals) is the Borel -algebra. The random variables X :(\Omega, \mathcal F, \operatorname P) \to \Reals and Y : (\tilde\Omega,\tilde, \tilde) \to \Reals (on two possibly different
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
s) are (or ), denoted by X \,\stackrel\, 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 \mathcal_X and \mathcal_Y agree on the -system \ which generates \mathcal(\Reals), and so by the example above: \mathcal_X = \mathcal_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, \mathcal, \operatorname), with respectively generated -systems \mathcal_X and \mathcal_Y. The joint cumulative distribution function of (X, Y) is F_(a,b) = \operatorname\left X \leq a,Y\leq b \right = \operatorname\left X^((-\infty, a \cap Y^((-\infty, b]) \right], \quad \text a, b \in \R. However, A = X^((-\infty, a]) \in \mathcal_X and B = Y^((-\infty,b]) \in \mathcal_Y. Because \mathcal_ = \left\ 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 process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
es, two processes (X_t)_, (Y_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. \left(X_, \ldots, X_\right) \,\stackrel\, \left(Y_, \ldots, Y_\right). The proof of this is another application of the - theorem.Kallenberg, ''Foundations Of Modern probability'', p. 48


Independent random variables

The theory of -system plays an important role in the probabilistic notion of
independence Independence is a condition of a person, nation, country, or state in which residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the st ...
. If X and Y are two random variables defined on the same probability space (\Omega, \mathcal, \operatorname) then the random variables are independent if and only if their -systems \mathcal_X, \mathcal_Y satisfy \operatorname \cap B= \operatorname \operatorname \quad \text A \in \mathcal_X \text B \in \mathcal_Y, which is to say that \mathcal_X, \mathcal_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 \mathcal(0, 1) are
iid In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usual ...
standard normal random variables. Define the radius and argument (arctan) variables R = \sqrt, \qquad \Theta = \tan^\left(Z_2 / Z_1\right). Then R and \Theta are independent random variables. To prove this, it is sufficient to show that the -systems \mathcal_R, \mathcal_\Theta are independent: that is, \operatorname \leq \rho, \Theta \leq \theta= \operatorname \leq \rho\operatorname Theta \leq \theta\quad \text \rho \in , \infty), \, \theta \in [0, 2 \pi Confirming that this is the case is an exercise in changing variables. Fix \rho \in [0, \infty) and \theta \in [0, 2 \pi], then the probability can be expressed as an integral of the probability density function of Z. \begin \operatorname P [ R \leq \rho, \Theta \leq \theta] &= \int_ \frac\exp\left(\right) dz_1 \, dz_2 \\ pt& = \int_0^ \int_0^\rho \frace^r \, dr \, d\tilde\theta \\ pt& = \left( \int_0^\theta \frac \, d\tilde \theta \right) \left( \int_0^\rho e^ r \, dr\right) \\ pt& = \operatorname P Theta \leq \thetaoperatorname P \leq \rho \end


See also

* * * * * * * * * * *


Notes


Citations


References

* * * {{DEFAULTSORT:Pi System Measure theory Families of sets