σ-algebra
Algebraic structure of set algebra / From Wikipedia, the free encyclopedia
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In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair is called a measurable space.
A σ-algebra of subsets is a set algebra of subsets; elements of the latter only need to be closed under the union or intersection of finitely many subsets, which is a weaker condition.[1]
The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events which can be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation.
In statistics, (sub) σ-algebras are needed for the formal mathematical definition of a sufficient statistic,[2] particularly when the statistic is a function or a random process and the notion of conditional density is not applicable.
If one possible σ-algebra on is where is the empty set. In general, a finite algebra is always a σ-algebra.
If is a countable partition of then the collection of all unions of sets in the partition (including the empty set) is a σ-algebra.
A more useful example is the set of subsets of the real line formed by starting with all open intervals and adding in all countable unions, countable intersections, and relative complements and continuing this process (by transfinite iteration through all countable ordinals) until the relevant closure properties are achieved (a construction known as the Borel hierarchy).
There are at least three key motivators for σ-algebras: defining measures, manipulating limits of sets, and managing partial information characterized by sets.
Measure
A measure on is a function that assigns a non-negative real number to subsets of this can be thought of as making precise a notion of "size" or "volume" for sets. We want the size of the union of disjoint sets to be the sum of their individual sizes, even for an infinite sequence of disjoint sets.
One would like to assign a size to every subset of but in many natural settings, this is not possible. For example, the axiom of choice implies that when the size under consideration is the ordinary notion of length for subsets of the real line, then there exist sets for which no size exists, for example, the Vitali sets. For this reason, one considers instead a smaller collection of privileged subsets of These subsets will be called the measurable sets. They are closed under operations that one would expect for measurable sets, that is, the complement of a measurable set is a measurable set and the countable union of measurable sets is a measurable set. Non-empty collections of sets with these properties are called σ-algebras.
Limits of sets
Many uses of measure, such as the probability concept of almost sure convergence, involve limits of sequences of sets. For this, closure under countable unions and intersections is paramount. Set limits are defined as follows on σ-algebras.
- The limit supremum or outer limit of a sequence of subsets of is It consists of all points that are in infinitely many of these sets (or equivalently, that are in cofinally many of them). That is, if and only if there exists an infinite subsequence (where ) of sets that all contain that is, such that
- The limit infimum or inner limit of a sequence of subsets of is It consists of all points that are in all but finitely many of these sets (or equivalently, that are eventually in all of them). That is, if and only if there exists an index such that all contain that is, such that
The inner limit is always a subset of the outer limit:
If these two sets are equal then their limit exists and is equal to this common set:
Sub σ-algebras
In much of probability, especially when conditional expectation is involved, one is concerned with sets that represent only part of all the possible information that can be observed. This partial information can be characterized with a smaller σ-algebra which is a subset of the principal σ-algebra; it consists of the collection of subsets relevant only to and determined only by the partial information. A simple example suffices to illustrate this idea.
Imagine you and another person are betting on a game that involves flipping a coin repeatedly and observing whether it comes up Heads () or Tails (). Since you and your opponent are each infinitely wealthy, there is no limit to how long the game can last. This means the sample space Ω must consist of all possible infinite sequences of or
However, after flips of the coin, you may want to determine or revise your betting strategy in advance of the next flip. The observed information at that point can be described in terms of the 2n possibilities for the first flips. Formally, since you need to use subsets of Ω, this is codified as the σ-algebra
Observe that then
where is the smallest σ-algebra containing all the others.
Definition
Let be some set, and let represent its power set. Then a subset is called a σ-algebra if and only if it satisfies the following three properties:[3]
- is in and is considered to be the universal set in the following context.
- is closed under complementation: If some set is in then so is its complement,
- is closed under countable unions: If are in then so is
From these properties, it follows that the σ-algebra is also closed under countable intersections (by applying De Morgan's laws).
It also follows that the empty set is in since by (1) is in and (2) asserts that its complement, the empty set, is also in Moreover, since satisfies condition (3) as well, it follows that is the smallest possible σ-algebra on The largest possible σ-algebra on is
Elements of the σ-algebra are called measurable sets. An ordered pair where is a set and is a σ-algebra over is called a measurable space. A function between two measurable spaces is called a measurable function if the preimage of every measurable set is measurable. The collection of measurable spaces forms a category, with the measurable functions as morphisms. Measures are defined as certain types of functions from a σ-algebra to
A σ-algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin's theorem (see below).
Dynkin's π-λ theorem
This theorem (or the related monotone class theorem) is an essential tool for proving many results about properties of specific σ-algebras. It capitalizes on the nature of two simpler classes of sets, namely the following.
- A π-system is a collection of subsets of that is closed under finitely many intersections, and
- A Dynkin system (or λ-system) is a collection of subsets of that contains and is closed under complement and under countable unions of disjoint subsets.
Dynkin's π-λ theorem says, if is a π-system and is a Dynkin system that contains then the σ-algebra generated by is contained in Since certain π-systems are relatively simple classes, it may not be hard to verify that all sets in enjoy the property under consideration while, on the other hand, showing that the collection of all subsets with the property is a Dynkin system can also be straightforward. Dynkin's π-λ Theorem then implies that all sets in enjoy the property, avoiding the task of checking it for an arbitrary set in
One of the most fundamental uses of the π-λ theorem is to show equivalence of separately defined measures or integrals. For example, it is used to equate a probability for a random variable with the Lebesgue-Stieltjes integral typically associated with computing the probability:
for all in the Borel σ-algebra on
where is the cumulative distribution function for defined on while is a probability measure, defined on a σ-algebra of subsets of some sample space
Combining σ-algebras
Suppose is a collection of σ-algebras on a space
Meet
The intersection of a collection of σ-algebras is a σ-algebra. To emphasize its character as a σ-algebra, it often is denoted by:
Sketch of Proof: Let denote the intersection. Since is in every is not empty. Closure under complement and countable unions for every implies the same must be true for Therefore, is a σ-algebra.
Join
The union of a collection of σ-algebras is not generally a σ-algebra, or even an algebra, but it generates a σ-algebra known as the join which typically is denoted
A π-system that generates the join is
Sketch of Proof: By the case it is seen that each so
This implies
by the definition of a σ-algebra generated by a collection of subsets. On the other hand,
which, by Dynkin's π-λ theorem, implies
σ-algebras for subspaces
Suppose is a subset of and let be a measurable space.
- The collection is a σ-algebra of subsets of
- Suppose is a measurable space. The collection is a σ-algebra of subsets of
Relation to σ-ring
A σ-algebra is just a σ-ring that contains the universal set [4] A σ-ring need not be a σ-algebra, as for example measurable subsets of zero Lebesgue measure in the real line are a σ-ring, but not a σ-algebra since the real line has infinite measure and thus cannot be obtained by their countable union. If, instead of zero measure, one takes measurable subsets of finite Lebesgue measure, those are a ring but not a σ-ring, since the real line can be obtained by their countable union yet its measure is not finite.
Typographic note
σ-algebras are sometimes denoted using calligraphic capital letters, or the Fraktur typeface. Thus may be denoted as or