The σ-algebras are a subset of the set algebras; elements of the latter only need to be closed under the union or intersection of finitely many subsets, which is a weaker condition.
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.
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.
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
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:
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.
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 (below).
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:
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.
A separable -algebra (or separable -field) is a -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 -algebra generated by a countable collection of sets is separable, but the converse need not hold. For example, the Lebesgue -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.
The collection is a simple σ-algebra generated by the subset
The collection of subsets of which 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.
A stopping time can define a -algebra the
so-called stopping time sigma-algebra, 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 
Let be an arbitrary family of subsets of Then there exists a unique smallest σ-algebra which 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
If is empty, then Otherwise consists of all the subsets of that can be made from elements of by a countable number of complement, union and intersection operations.
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, may be written instead 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.
If is a function from to then is generated by the family of subsets which 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.
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
the σ-algebra generated by the inverse images of cylinder sets.