In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection of subsets of X that includes X itself, is closed under complement, and is closed under countable unions.
The pair is called a measurable space or Borel space.
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 that 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, 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 - the σ-algebra produced by this process is known as the Borel algebra on the real line, and can also be conceived as the smallest (i.e. "coarsest") σ-algebra containing all the open sets, or equivalently containing all the closed sets. It is foundational to measure theory, and therefore modern probability theory, and a related construction known as the Borel hierarchy is of relevance to descriptive set theory.
There are at least three key motivators for σ-algebras: defining measures, manipulating limits of sets, and managing partial information characterized by sets.
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 setsEdit
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 of a sequence each of which is a subset of is
- The limit infimum of a sequence each of which is a subset of is
- If, in fact,
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 that 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 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 this is codified as the σ-algebra
Observe that then
Definition and propertiesEdit
- is closed under complementation in : If is an element of then so is its complement
- In this context, is considered to be the universal set.
- contains as an element:
- Assuming that (1) holds, this condition is equivalent to containing the empty set:
- is closed under countable unions: If are elements of then so is their union
Equivalently, a σ-algebra is an algebra of sets that is closed under countable unions.
The empty set belongs to because by (1), is in and so (2) implies 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
Dynkin's π-λ theoremEdit
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
Suppose that is a collection of σ-algebras on a space
- 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.
- 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
Sketch of Proof
By the case it is seen that each so
σ-algebras for subspacesEdit
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-ringEdit
A σ-algebra is just a σ-ring that contains the universal set  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.
Particular cases and examplesEdit
A separable σ-algebra (or separable σ-field) is a σ-algebra that is a separable space when considered as a metric space with metric for and a given measure (and with being the symmetric difference operator). Note that any σ-algebra generated by a countable collection of sets is separable, but the converse need not hold. For example, the Lebesgue σ-algebra is separable (since every Lebesgue measurable set is equivalent to some Borel set) but not countably generated (since its cardinality is higher than continuum).
A separable measure space has a natural pseudometric that renders it separable as a pseudometric space. The distance between two sets is defined as the measure of the symmetric difference of the two sets. Note that the symmetric difference of two distinct sets can have measure zero; hence the pseudometric as defined above need not to be a true metric. However, if sets whose symmetric difference has measure zero are identified into a single equivalence class, the resulting quotient set can be properly metrized by the induced metric. If the measure space is separable, it can be shown that the corresponding metric space is, too.
Simple set-based examplesEdit
Let be any set.
- The family consisting only of the empty set and the set called the minimal or trivial σ-algebra over
- The power set of called the discrete σ-algebra.
- The collection is a simple σ-algebra generated by the subset
- The collection of subsets of that are countable or whose complements are countable is a σ-algebra (which is distinct from the power set of if and only if is uncountable). This is the σ-algebra generated by the singletons of Note: "countable" includes finite or empty.
- The collection of all unions of sets in a countable partition of is a σ-algebra.
Stopping time σ-algebrasEdit
A stopping time can define a -algebra , the so-called -Algebra of τ-past, which in a filtered probability space describes the information up to the random time in the sense that, if the filtered probability space is interpreted as a random experiment, the maximum information that can be found out about the experiment from arbitrarily often repeating it until the time is .
σ-algebras generated by families of setsEdit
σ-algebra generated by an arbitrary familyEdit
Let be an arbitrary family of subsets of Then there exists a unique smallest σ-algebra that contains every set in (even though may or may not itself be a σ-algebra). It is, in fact, the intersection of all σ-algebras containing (See intersections of σ-algebras above.) This σ-algebra is denoted and is called the σ-algebra generated by
Then consists of all the subsets of that can be made from elements of by a countable number of complement, union and intersection operations. If is empty, then since an empty union and intersection produce the empty set and universal set, respectively.
For a simple example, consider the set Then the σ-algebra generated by the single subset is By an abuse of notation, when a collection of subsets contains only one element, one may write instead of if it is clear that is a subset of ; in the prior example instead of Indeed, using to mean is also quite common.
There are many families of subsets that generate useful σ-algebras. Some of these are presented here.
σ-algebra generated by a functionEdit
If is a function from a set to a set and is a σ-algebra of subsets of then the σ-algebra generated by the function denoted by is the collection of all inverse images of the sets in that is,
A function from a set to a set is measurable with respect to a σ-algebra of subsets of if and only if is a subset of
If is a function from to then is generated by the family of subsets that are inverse images of intervals/rectangles in :
A useful property is the following. Assume is a measurable map from to and is a measurable map from to If there exists a measurable map from to such that for all then If is finite or countably infinite or, more generally, is a standard Borel space (for example, a separable complete metric space with its associated Borel sets), then the converse is also true. Examples of standard Borel spaces include with its Borel sets and with the cylinder σ-algebra described below.
Borel and Lebesgue σ-algebrasEdit
An important example is the Borel algebra over any topological space: the σ-algebra generated by the open sets (or, equivalently, by the closed sets). Note that this σ-algebra is not, in general, the whole power set. For a non-trivial example that is not a Borel set, see the Vitali set or Non-Borel sets.
On the Euclidean space another σ-algebra is of importance: that of all Lebesgue measurable sets. This σ-algebra contains more sets than the Borel σ-algebra on and is preferred in integration theory, as it gives a complete measure space.
Let and be two measurable spaces. The σ-algebra for the corresponding product space is called the product σ-algebra and is defined by
Observe that is a π-system.
The Borel σ-algebra for is generated by half-infinite rectangles and by finite rectangles. For example,
For each of these two examples, the generating family is a π-system.
σ-algebra generated by cylinder setsEdit
An important special case is when is the set of natural numbers and is a set of real-valued sequences. In this case, it suffices to consider the cylinder sets
σ-algebra generated by random variable or vectorEdit
σ-algebra generated by a stochastic processEdit
Suppose is a probability space and is the set of real-valued functions on . If is measurable with respect to the cylinder σ-algebra (see above) for then is called a stochastic process or random process. The σ-algebra generated by is
- Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets.
- Join (sigma algebra)
- 𝜆-system (Dynkin system)
- Measurable function – function for which the preimage of a measurable set is measurable
- π-system – Non-empty family of sets where the intersection of any two members is again a member
- Ring of sets
- Sample space
- 𝜎 additivity
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If is a Borel measure on , the measure algebra of is the Boolean algebra of all Borel sets modulo -null sets. If is finite, then such a measure algebra is also a metric space, with the distance between the two sets being the measure of their symmetric difference. Then, we say that is separable if and only if this metric space is separable as a topological space.
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