Sigmaalgebra
In mathematical analysis and in probability theory, a σalgebra (also sigmaalgebra, σfield, sigmafield) on a set X is a collection Σ of subsets of X that is closed under countablefold set operations (complement, union of countably many sets and intersection of countably many sets). By contrast, an algebra is only required to be closed under finitely many set operations. That is, a σalgebra is an algebra of sets, completed to include countably infinite operations. The pair (X, Σ) is also a field of sets, called a measurable space.
The main use of σalgebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σalgebra. 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, σalgebras are pivotal in the definition of conditional expectation.
In statistics, (sub) σalgebras are needed for a formal mathematical definition of sufficient statistic,^{[1]} particularly when the statistic is a function or a random process and the notion of conditional density is not applicable.
If X = {a, b, c, d}, one possible σalgebra on X is Σ = { ∅, {a, b}, {c, d}, {a, b, c, d} }, where ∅ is the empty set. However, a finite algebra is always a σalgebra.
If {A_{1}, A_{2}, A_{3}, …} is a countable partition of X then the collection of all unions of sets in the partition (including the empty set) is a σalgebra.
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).
Contents
Motivation
There are at least three key motivators for σalgebras: defining measures, manipulating limits of sets, and managing partial information characterized by sets.
Measure
A measure on X is a function that assigns a nonnegative real number to subsets of X; 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 X, 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 X. 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. Nonempty collections of sets with these properties are called σalgebras.
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 σalgebras.
 The limit supremum of a sequence A_{1}, A_{2}, A_{3}, ..., each of which is a subset of X, is
 The limit infimum of a sequence A_{1}, A_{2}, A_{3}, ..., each of which is a subset of X, is
 If, in fact,

 then the exists as that common set.
Sub σalgebras
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 σalgebra which is a subset of the principal σalgebra; 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 are playing a game that involves flipping a coin repeatedly and observing whether it comes up Heads (H) or Tails (T). 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 H or T:
 .
However, after n 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 2^{n} possibilities for the first n flips. Formally, since you need to use subsets of Ω, this is codified as the σalgebra
 .
Observe that then
 ,
where is the smallest σalgebra containing all the others.
Definition and properties
Definition
Let X be some set, and let 2^{X} represent its power set. Then a subset Σ ⊂ 2^{X} is called a σalgebra if it satisfies the following three properties:^{[2]}
 X is in Σ, and X is considered to be the universal set in the following context.
 Σ is closed under complementation: If A is in Σ, then so is its complement, X\A.
 Σ is closed under countable unions: If A_{1}, A_{2}, A_{3}, ... are in Σ, then so is A = A_{1} ∪ A_{2} ∪ A_{3} ∪ … .
From these properties, it follows that the σalgebra is also closed under countable intersections (by applying De Morgan's laws).
It also follows that the empty set ∅ is in Σ, since by (1) X is in Σ and (2) asserts that its complement, the empty set, is also in Σ. Moreover, by (3) it follows as well that {X, ∅} is the smallest possible σalgebra.
Elements of the σalgebra are called measurable sets. An ordered pair (X, Σ), where X is a set and Σ is a σalgebra over X, 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 σalgebra to [0, ∞].
A σalgebra is both a πsystem and a Dynkin system (λsystem). The converse is true as well, by Dynkin's theorem (below).
Dynkin's πλ theorem
This theorem (or the related monotone class theorem) is an essential tool for proving many results about properties of specific σalgebras. It capitalizes on the nature of two simpler classes of sets, namely the following.
 A πsystem P is a collection of subsets of Σ that is closed under finitely many intersections, and
 a Dynkin system (or λsystem) D is a collection of subsets of Σ that contains Σ and is closed under complement and under countable unions of disjoint subsets.
Dynkin's πλ theorem says, if P is a πsystem and D is a Dynkin system that contains P then the σalgebra σ(P) generated by P is contained in D. Since certain πsystems are relatively simple classes, it may not be hard to verify that all sets in P enjoy the property under consideration while, on the other hand, showing that the collection D of all subsets with the property is a Dynkin system can also be straightforward. Dynkin's πλ Theorem then implies that all sets in σ(P) enjoy the property, avoiding the task of checking it for an arbitrary set in σ(P).
One of the most fundamental uses of the πλ theorem is to show equivalence of separately defined measures or integrals. For example, it is used to equate a probability for a random variable X with the LebesgueStieltjes integral typically associated with computing the probability:
 for all A in the Borel σalgebra on R,
where F(x) is the cumulative distribution function for X, defined on R, while is a probability measure, defined on a σalgebra Σ of subsets of some sample space Ω.
Combining σalgebras
Suppose is a collection of σalgebras on a space X.
 The intersection of a collection of σalgebras is a σalgebra. To emphasize its character as a σalgebra, it often is denoted by:

 Sketch of Proof: Let Σ^{∗} denote the intersection. Since X is in every Σ_{α}, Σ^{∗} is not empty. Closure under complement and countable unions for every Σ_{α} implies the same must be true for Σ^{∗}. Therefore, Σ^{∗} is a σalgebra.
 The union of a collection of σalgebras is not generally a σalgebra, or even an algebra, but it generates a σalgebra known as the join which typically is denoted

 A πsystem that generates the join is
 Sketch of Proof: By the case n = 1, it is seen that each , so
 This implies
 by the definition of a σalgebra generated by a collection of subsets. On the other hand,
 which, by Dynkin's πλ theorem, implies
σalgebras for subspaces
Suppose Y is a subset of X and let (X, Σ) be a measurable space.
 The collection {Y ∩ B: B ∈ Σ} is a σalgebra of subsets of Y.
 Suppose (Y, Λ) is a measurable space. The collection {A ⊂ X : A ∩ Y ∈ Λ} is a σalgebra of subsets of X.
Relation to σring
A σalgebra Σ is just a σring that contains the universal set X.^{[3]} A σring need not be a σalgebra, as for example measurable subsets of zero Lebesgue measure in the real line are a σring, but not a σalgebra 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 σring, since the real line can be obtained by their countable union yet its measure is not finite.
Typographic note
σalgebras are sometimes denoted using calligraphic capital letters, or the Fraktur typeface. Thus (X, Σ) may be denoted as or .
Examples
Simple setbased examples
Let X be any set.
 The family consisting only of the empty set and the set X, called the minimal or trivial σalgebra over X.
 The power set of X, called the discrete σalgebra.
 The collection {∅, A, A^{c}, X} is a simple σalgebra generated by the subset A.
 The collection of subsets of X which are countable or whose complements are countable is a σalgebra (which is distinct from the power set of X if and only if X is uncountable). This is the σalgebra generated by the singletons of X. Note: "countable" includes finite or empty.
 The collection of all unions of sets in a countable partition of X is a σalgebra.
Stopping time sigmaalgebras
A stopping time can define a algebra , the socalled stopping time sigmaalgebra, 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 .^{[4]}
σalgebras generated by families of sets
σalgebra generated by an arbitrary family
Let F be an arbitrary family of subsets of X. Then there exists a unique smallest σalgebra which contains every set in F (even though F may or may not itself be a σalgebra). It is, in fact, the intersection of all σalgebras containing F. (See intersections of σalgebras above.) This σalgebra is denoted σ(F) and is called the σalgebra generated by F.
For a simple example, consider the set X = {1, 2, 3}. Then the σalgebra generated by the single subset {1} is σ({{1}}) = {∅, {1}, {2, 3}, {1, 2, 3}}. By an abuse of notation, when a collection of subsets contains only one element, A, one may write σ(A) instead of σ({A}); in the prior example σ({1}) instead of σ({{1}}). Indeed, using σ(A_{1}, A_{2}, ...) to mean σ({A_{1}, A_{2}, ...}) is also quite common.
There are many families of subsets that generate useful σalgebras. Some of these are presented here.
σalgebra generated by a function
If f is a function from a set X to a set Y and B is a σalgebra of subsets of Y, then the σalgebra generated by the function f, denoted by σ(f), is the collection of all inverse images f^{−1}(S) of the sets S in B. i.e.
A function f from a set X to a set Y is measurable with respect to a σalgebra Σ of subsets of X if and only if σ(f) is a subset of Σ.
One common situation, and understood by default if B is not specified explicitly, is when Y is a metric or topological space and B is the collection of Borel sets on Y.
If f is a function from X to R^{n} then σ(f) is generated by the family of subsets which are inverse images of intervals/rectangles in R^{n}:
A useful property is the following. Assume f is a measurable map from (X, Σ_{X}) to (S, Σ_{S}) and g is a measurable map from (X, Σ_{X}) to (T, Σ_{T}). If there exists a measurable function h from T to S such that f(x) = h(g(x)) then σ(f) ⊂ σ(g). If S is finite or countably infinite or if (S, Σ_{S}) is a standard Borel space (e.g., a separable complete metric space with its associated Borel sets) then the converse is also true.^{[5]} Examples of standard Borel spaces include R^{n} with its Borel sets and R^{∞} with the cylinder σalgebra described below.
Borel and Lebesgue σalgebras
An important example is the Borel algebra over any topological space: the σalgebra generated by the open sets (or, equivalently, by the closed sets). Note that this σalgebra is not, in general, the whole power set. For a nontrivial example that is not a Borel set, see the Vitali set or NonBorel sets.
On the Euclidean space R^{n}, another σalgebra is of importance: that of all Lebesgue measurable sets. This σalgebra contains more sets than the Borel σalgebra on R^{n} and is preferred in integration theory, as it gives a complete measure space.
Product σalgebra
Let and be two measurable spaces. The σalgebra for the corresponding product space is called the product σalgebra and is defined by
Observe that is a πsystem.
The Borel σalgebra for R^{n} is generated by halfinfinite rectangles and by finite rectangles. For example,
For each of these two examples, the generating family is a πsystem.
σalgebra generated by cylinder sets
Suppose
is a set of realvalued functions. Let denote the Borel subsets of R. A cylinder subset of X is a finitely restricted set defined as
Each
is a πsystem that generates a σalgebra . Then the family of subsets
is an algebra that generates the cylinder σalgebra for X. This σalgebra is a subalgebra of the Borel σalgebra determined by the product topology of restricted to X.
An important special case is when is the set of natural numbers and X is a set of realvalued sequences. In this case, it suffices to consider the cylinder sets
for which
is a nondecreasing sequence of σalgebras.
σalgebra generated by random variable or vector
Suppose is a probability space. If is measurable with respect to the Borel σalgebra on R^{n} then Y is called a random variable (n = 1) or random vector (n ≥ 1). The σalgebra generated by Y is
σalgebra generated by a stochastic process
Suppose is a probability space and is the set of realvalued functions on . If is measurable with respect to the cylinder σalgebra (see above) for X then Y is called a stochastic process or random process. The σalgebra generated by Y is
the σalgebra generated by the inverse images of cylinder sets.
See also
 Join (sigma algebra)
 Measurable function
 Sample space
 Separable sigma algebra
 Sigma ring
 Sigma additivity
References
 ↑ Billingsley, Patrick (2012). Probability and Measure (Anniversary ed.). Wiley. ISBN 9781118122372.
 ↑ Rudin, Walter (1987). Real & Complex Analysis. McGrawHill. ISBN 0070542341.
 ↑ Vestrup, Eric M. (2009). The Theory of Measures and Integration. John Wiley & Sons. p. 12. ISBN 9780470317952.
 ↑ Fischer, Tom (2013). "On simple representations of stopping times and stopping time sigmaalgebras". Statistics and Probability Letters. 83 (1): 345–349. doi:10.1016/j.spl.2012.09.024.
 ↑ Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). Springer. p. 7. ISBN 0387953132.
External links
 Hazewinkel, Michiel, ed. (2001), "Algebra of sets", Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 Sigma Algebra from PlanetMath.