Σ-algebra

In mathematical analysis and in probability theory, a σ-algebra is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with area or volume. In probability theory, they are used to define events for which a probability can be defined. In this way, σ-algebras help to formalize the notion of size.
In formal terms, a σ-algebra on a set is a nonempty collection of subsets of closed under complement, countable unions, and countable intersections. The ordered pair is called a measurable space.
The set is understood to be an ambient space, and the collection is a choice of subsets declared to have a well-defined size. The closure requirements for σ-algebras are designed to capture our intuitive ideas about how sizes combine: if there is a well-defined probability that an event occurs, there should be a well-defined probability that it does not occur ; if several sets have a well-defined size, so should their combination ; if several events have a well-defined probability of occurring, so should the event where they all occur simultaneously.
The definition of σ-algebra resembles other mathematical structures such as a topology or a set algebra.

Examples of σ-algebras

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 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 until the relevant closure properties are achieved.

Motivation

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 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 or of a sequence of subsets of is It consists of all points that are in infinitely many of these sets. That is, if and only if there exists an infinite subsequence of sets that all contain that is, such that
The or of a sequence of subsets of is It consists of all points that are in all but finitely many of these sets. 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. Formally, if are σ-algebras on, then is a sub σ-algebra of if.
The Bernoulli process provides a simple example. This consists of a sequence of random coin flips, coming up Heads or Tails, of unbounded length. The sample space Ω consists of all possible infinite sequences of or
The full sigma algebra can be generated from an ascending sequence of subalgebras, by considering the information that might be obtained after observing some or all of the first coin flips. This sequence of subalgebras is given by
Each of these is finer than the last, and so can be ordered as a filtration
The first subalgebra is the trivial algebra: it has only two elements in it, the empty set and the total space. The second subalgebra has four elements: the two in plus two more: sequences that start with and sequences that start with. Each subalgebra is finer than the last. The 'th subalgebra contains elements: it divides the total space into all of the possible sequences that might have been observed after flips, including the possible non-observation of some of the flips.
The limiting algebra is the smallest σ-algebra containing all the others. It is the algebra generated by the product topology or weak topology on the product space

Definition and properties

Definition

Let be some set, and let represent its power set, the set of all subsets of. Then a subset is called a σ-algebra if it satisfies the following three properties:

is in.
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.
It also follows that the empty set is in since by ' is in and ' asserts that its complement, the empty set, is also in Moreover, since satisfies all 3 conditions, 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. The converse is true as well, by Dynkin's theorem.

Dynkin's π-λ theorem

This 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 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 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.