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In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection ${\displaystyle \Sigma }$ of subsets of X satisfying the following conditions : (1) it includes X itself, (2) it is closed under complement, (3) it is closed under countable unions, and (4) it is closed under countable intersections. Note that (1) and (2) implies that it includes the empty subset.

The pair ${\displaystyle (X,\Sigma )}$ is called a measurable space or Borel space.

A σ-algebra is a type of algebra of sets. An algebra of sets needs only 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 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 ${\displaystyle X=\{a,b,c,d\},}$ one possible σ-algebra on ${\displaystyle X}$ is ${\displaystyle \Sigma =\{\varnothing ,\{a,b\},\{c,d\},\{a,b,c,d\}\},}$ where ${\displaystyle \varnothing }$ is the empty set. In general, a finite algebra is always a σ-algebra.

If ${\displaystyle \left\{A_{1},A_{2},A_{3},\ldots \right\}}$ is a countable partition of ${\displaystyle X}$ 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.

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 ${\displaystyle X}$ is a function that assigns a non-negative real number to subsets of ${\displaystyle X}$; 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 ${\displaystyle X,}$ 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 ${\displaystyle X.}$ 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. 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 of a sequence ${\displaystyle A_{1},A_{2},A_{3},\ldots ,}$ each of which is a subset of ${\displaystyle X,}$ is $${\displaystyle \limsup _{n\to \infty }A_{n}=\bigcap _{n=1}^{\infty }\bigcup _{m=n}^{\infty }A_{m}.}$$
• The limit infimum of a sequence ${\displaystyle A_{1},A_{2},A_{3},\ldots ,}$ each of which is a subset of ${\displaystyle X,}$ is $${\displaystyle \liminf _{n\to \infty }A_{n}=\bigcup _{n=1}^{\infty }\bigcap _{m=n}^{\infty }A_{m}.}$$
• If, in fact, $${\displaystyle \liminf _{n\to \infty }A_{n}=\limsup _{n\to \infty }A_{n},}$$ then the ${\textstyle \lim _{n\to \infty }A_{n}}$ exists as that 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 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 (${\displaystyle H}$) or Tails (${\displaystyle T}$). 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 ${\displaystyle \Omega }$ must consist of all possible infinite sequences of ${\displaystyle H}$ or ${\displaystyle T}$: $${\displaystyle \Omega =\{H,T\}^{\infty }=\{(x_{1},x_{2},x_{3},\dots ):x_{i}\in \{H,T\},i\geq 1\}.}$$

However, after ${\displaystyle n}$ 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 possibilities for the first ${\displaystyle n}$ flips. Formally, since you need to use subsets of ${\displaystyle \Omega ,}$ this is codified as the σ-algebra $${\displaystyle {\mathcal {G}}_{n}=\{A\times \{H,T\}^{\infty }:A\subseteq \{H,T\}^{n}\}.}$$

Observe that then $${\displaystyle {\mathcal {G}}_{1}\subseteq {\mathcal {G}}_{2}\subseteq {\mathcal {G}}_{3}\subseteq \cdots \subseteq {\mathcal {G}}_{\infty },}$$ where ${\displaystyle {\mathcal {G}}_{\infty }}$ is the smallest σ-algebra containing all the others.

Definition and properties

Definition

Throughout, ${\displaystyle X}$ will be a set and ${\displaystyle {\mathcal {P}}(X)}$ will denote its power set. A subset ${\displaystyle \Sigma \subseteq {\mathcal {P}}(X)}$ is called a σ-algebra if it has the following three properties:

1. ${\displaystyle \Sigma }$ is closed under complementation in ${\displaystyle X}$: If ${\displaystyle S}$ is an element of ${\displaystyle \Sigma }$ then so is its complement ${\displaystyle X\setminus S.}$
   • In this context, ${\displaystyle X}$ is considered to be the universal set.
2. ${\displaystyle \Sigma }$ contains ${\displaystyle X}$ as an element: ${\displaystyle X\in \Sigma .}$
   • Assuming that (1) holds, this condition is equivalent to ${\displaystyle \Sigma }$ containing the empty set: ${\displaystyle \varnothing \in \Sigma .}$
3. ${\displaystyle \Sigma }$ is closed under countable unions: If ${\displaystyle S_{1},S_{2},S_{3},\ldots }$ are elements of ${\displaystyle \Sigma }$ then so is their union ${\textstyle \bigcup _{i=1}^{\infty }S_{i}:=S_{1}\cup S_{2}\cup S_{3}\cup \cdots .}$
   • Assuming that (1) and (2) hold, it follows from De Morgan's laws that this condition is equivalent to ${\displaystyle \Sigma }$ being closed under countable intersections: If ${\displaystyle S_{1},S_{2},S_{3},\ldots }$ are elements of ${\displaystyle \Sigma }$ then so is their intersection ${\textstyle \bigcap _{i=1}^{\infty }S_{i}:=S_{1}\cap S_{2}\cap S_{3}\cap \cdots .}$

Equivalently, a σ-algebra is an algebra of sets that is closed under countable unions.

The empty set ${\displaystyle \varnothing }$ belongs to ${\displaystyle \Sigma }$ because by (2), ${\displaystyle X}$ is in ${\displaystyle \Sigma }$ and so (1) implies that its complement, the empty set, is also in ${\displaystyle \Sigma .}$ Moreover, since ${\displaystyle \{X,\varnothing \}}$ satisfies condition (3) as well, it follows that ${\displaystyle \{X,\varnothing \}}$ is the smallest possible σ-algebra on ${\displaystyle X.}$ The largest possible σ-algebra on ${\displaystyle X}$ is ${\displaystyle 2^{X}:={\mathcal {P}}(X).}$

Elements of the σ-algebra are called measurable sets. An ordered pair ${\displaystyle (X,\Sigma ),}$ where ${\displaystyle X}$ is a set and ${\displaystyle \Sigma }$ is a σ-algebra over ${\displaystyle X,}$ 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 σ-algebra to ${\displaystyle .}$

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).

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 ${\displaystyle P}$ is a collection of subsets of ${\displaystyle X}$ that is closed under finitely many intersections, and

a Dynkin system (or 𝜆-system) ${\displaystyle D}$ is a collection of subsets of ${\displaystyle X}$ that contains ${\displaystyle X}$ and is closed under complement and under countable unions of disjoint subsets.

Dynkin's π-𝜆 theorem says, if ${\displaystyle P}$ is a π-system and ${\displaystyle D}$ is a Dynkin system that contains ${\displaystyle P}$ then the σ-algebra ${\displaystyle \sigma (P)}$ generated by ${\displaystyle P}$ is contained in ${\displaystyle D.}$ Since certain π-systems are relatively simple classes, it may not be hard to verify that all sets in ${\displaystyle P}$ enjoy the property under consideration while, on the other hand, showing that the collection ${\displaystyle D}$ 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 ${\displaystyle \sigma (P)}$ enjoy the property, avoiding the task of checking it for an arbitrary set in ${\displaystyle \sigma (P).}$

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 ${\displaystyle X}$ with the Lebesgue-Stieltjes integral typically associated with computing the probability:

${\displaystyle \mathbb {P} (X\in A)=\int _{A}\,F(dx)}$ for all ${\displaystyle A}$ in the Borel σ-algebra on ${\displaystyle \mathbb {R} ,}$

where ${\displaystyle F(x)}$ is the cumulative distribution function for ${\displaystyle X,}$ defined on ${\displaystyle \mathbb {R} ,}$ while ${\displaystyle \mathbb {P} }$ is a probability measure, defined on a σ-algebra ${\displaystyle \Sigma }$ of subsets of some sample space ${\displaystyle \Omega .}$

Combining σ-algebras

Suppose that ${\textstyle \{\Sigma _{\alpha }:\alpha \in {\mathcal {A}}\}}$ is a collection of σ-algebras on a space ${\displaystyle X.}$

• 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: $${\displaystyle \bigwedge _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }.}$$

  Sketch of Proof
  Let ${\displaystyle \Sigma ^{*}}$ denote the intersection. Since ${\displaystyle X}$ is in every ${\displaystyle \Sigma _{\alpha },\Sigma ^{*}}$ is not empty. Closure under complement and countable unions for every ${\displaystyle \Sigma _{\alpha }}$ implies the same must be true for ${\displaystyle \Sigma ^{*}.}$ Therefore, ${\displaystyle \Sigma ^{*}}$ 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 $${\displaystyle \bigvee _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }=\sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right).}$$ A π-system that generates the join is $${\displaystyle {\mathcal {P}}=\left\{\bigcap _{i=1}^{n}A_{i}:A_{i}\in \Sigma _{\alpha _{i}},\alpha _{i}\in {\mathcal {A}},\ n\geq 1\right\}.}$$

  Sketch of Proof

  By the case ${\displaystyle n=1,}$ it is seen that each ${\displaystyle \Sigma _{\alpha }\subseteq {\mathcal {P}},}$ so $${\displaystyle \bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\subseteq {\mathcal {P}}.}$$ This implies $${\displaystyle \sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right)\subseteq \sigma ({\mathcal {P}})}$$ by the definition of a σ-algebra generated by a collection of subsets. On the other hand, $${\displaystyle {\mathcal {P}}\subseteq \sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right)}$$ which, by Dynkin's π-𝜆 theorem, implies $${\displaystyle \sigma ({\mathcal {P}})\subseteq \sigma \left(\bigcup _{\alpha \in {\mathcal {A}}}\Sigma _{\alpha }\right).}$$

σ-algebras for subspaces

Suppose ${\displaystyle Y}$ is a subset of ${\displaystyle X}$ and let ${\displaystyle (X,\Sigma )}$ be a measurable space.

• The collection ${\displaystyle \{Y\cap S:S\in \Sigma \}}$ is a σ-algebra of subsets of ${\displaystyle Y.}$
• Suppose ${\displaystyle (Y,\Lambda )}$ is a measurable space. The collection ${\displaystyle \{A\subseteq X:A\cap Y\in \Lambda \}}$ is a σ-algebra of subsets of ${\displaystyle X.}$

Relation to σ-ring

A σ-algebra ${\displaystyle \Sigma }$ is just a σ-ring that contains the universal set ${\displaystyle X.}$ 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 ${\displaystyle (X,\Sigma )}$ may be denoted as ${\displaystyle (X,\,{\mathcal {F}})}$ or ${\displaystyle (X,\,{\mathfrak {F}}).}$

Particular cases and examples

Separable σ-algebras

A separable σ-algebra (or separable σ-field) is a σ-algebra ${\displaystyle {\mathcal {F}}}$ that is a separable space when considered as a metric space with metric ${\displaystyle \rho (A,B)=\mu (A{\mathbin {\triangle }}B)}$ for ${\displaystyle A,B\in {\mathcal {F}}}$ and a given measure ${\displaystyle \mu }$ (and with ${\displaystyle \triangle }$ 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 examples

Let ${\displaystyle X}$ be any set.

• The family consisting only of the empty set and the set ${\displaystyle X,}$ called the minimal or trivial σ-algebra over ${\displaystyle X.}$
• The power set of ${\displaystyle X,}$ called the discrete σ-algebra.
• The collection ${\displaystyle \left\{\varnothing ,X,A,X\setminus A\right\}}$ is a simple σ-algebra generated by the subset ${\displaystyle A.}$
• The collection of subsets of ${\displaystyle X}$ that are countable or whose complements are countable is a σ-algebra (which is distinct from the power set of ${\displaystyle X}$ if and only if ${\displaystyle X}$ is uncountable). This is the σ-algebra generated by the singletons of ${\displaystyle X.}$ Note: "countable" includes finite or empty.
• The collection of all unions of sets in a countable partition of ${\displaystyle X}$ is a σ-algebra. This remains true when unions are replaced by countable unions.

Stopping time σ-algebras

A stopping time ${\displaystyle \tau }$ can define a ${\displaystyle \sigma }$-algebra ${\displaystyle {\mathcal {F}}_{\tau }}$, the so-called ${\displaystyle \sigma }$-Algebra of τ-past, which in a filtered probability space describes the information up to the random time ${\displaystyle \tau }$ 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 ${\displaystyle \tau }$ is ${\displaystyle {\mathcal {F}}_{\tau }}$.

σ-algebras generated by families of sets

σ-algebra generated by an arbitrary family

Let ${\displaystyle F}$ be an arbitrary family of subsets of ${\displaystyle X.}$ Then there exists a unique smallest σ-algebra that contains every set in ${\displaystyle F}$ (even though ${\displaystyle F}$ may or may not itself be a σ-algebra). It is, in fact, the intersection of all σ-algebras containing ${\displaystyle F.}$ (See intersections of σ-algebras above.) This σ-algebra is denoted ${\displaystyle \sigma (F)}$ and is called the σ-algebra generated by ${\displaystyle F.}$

Then ${\displaystyle \sigma (F)}$ consists of all the subsets of ${\displaystyle X}$ that can be made from elements of ${\displaystyle F}$ by a countable number of complement, union and intersection operations. If ${\displaystyle F}$ is empty, then ${\displaystyle \sigma (F)=\{X,\varnothing \},}$ since an empty union and intersection produce the empty set and universal set, respectively.

For a simple example, consider the set ${\displaystyle X=\{1,2,3\}.}$ Then the σ-algebra generated by the single subset ${\displaystyle \{1\}}$ is ${\displaystyle \sigma (\{\{1\}\})=\{\varnothing ,\{1\},\{2,3\},\{1,2,3\}\}.}$ By an abuse of notation, when a collection of subsets contains only one element, ${\displaystyle A,}$ one may write ${\displaystyle \sigma (A)}$ instead of ${\displaystyle \sigma (\{A\})}$ if it is clear that ${\displaystyle A}$ is a subset of ${\displaystyle X}$; in the prior example ${\displaystyle \sigma (\{1\})}$ instead of ${\displaystyle \sigma (\{\{1\}\}).}$ Indeed, using ${\displaystyle \sigma \left(A_{1},A_{2},\ldots \right)}$ to mean ${\displaystyle \sigma \left(\left\{A_{1},A_{2},\ldots \right\}\right)}$ 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 function

If ${\displaystyle f:X\to Y}$ is a function from a set ${\displaystyle X}$ to a set ${\displaystyle Y}$ and ${\displaystyle B}$ is a σ-algebra of subsets of ${\displaystyle Y,}$ then the σ-algebra generated by the function ${\displaystyle f,}$ denoted by ${\displaystyle \sigma (f),}$ is the collection of all inverse images ${\displaystyle f^{-1}(S)}$ of the sets ${\displaystyle S}$ in ${\displaystyle B.}$ that is, $${\displaystyle \sigma (f)=\{f^{-1}(S):S\in B\}.}$$

A function ${\displaystyle f:X\to Y}$ from a set ${\displaystyle X}$ to a set ${\displaystyle Y}$ is measurable with respect to a σ-algebra ${\displaystyle \Sigma }$ of subsets of ${\displaystyle X}$ if and only if ${\displaystyle \sigma (f)}$ is a subset of ${\displaystyle \Sigma .}$

One common situation, and understood by default if ${\displaystyle B}$ is not specified explicitly, is when ${\displaystyle Y}$ is a metric or topological space and ${\displaystyle B}$ is the collection of Borel sets on ${\displaystyle Y.}$

If ${\displaystyle f:X\to \mathbb {R} ^{n}}$ is a function from ${\displaystyle X}$ to ${\displaystyle \mathbb {R} ^{n}}$ then ${\displaystyle \sigma (f)}$ is generated by the family of subsets that are inverse images of intervals/rectangles in ${\displaystyle \mathbb {R} ^{n}}$: $${\displaystyle \sigma (f)=\sigma \left(\{f^{-1}((a_{1},b_{1}]\times \cdots \times (a_{n},b_{n}]):a_{i},b_{i}\in \mathbb {R} \}\right).}$$

A useful property is the following. Assume ${\displaystyle f:X\to S}$ is a measurable map from ${\displaystyle \left(X,\Sigma _{X}\right)}$ to ${\displaystyle \left(S,\Sigma _{S}\right)}$ and ${\displaystyle g:X\to T}$ is a measurable map from ${\displaystyle \left(X,\Sigma _{X}\right)}$ to ${\displaystyle \left(T,\Sigma _{T}\right).}$ If there exists a measurable map ${\displaystyle h:T\to S}$ from ${\displaystyle \left(T,\Sigma _{T}\right)}$ to ${\displaystyle \left(S,\Sigma _{S}\right)}$ such that ${\displaystyle f(x)=h(g(x))}$ for all ${\displaystyle x,}$ then ${\displaystyle \sigma (f)\subseteq \sigma (g).}$ If ${\displaystyle S}$ is finite or countably infinite or, more generally, ${\displaystyle \left(S,\Sigma _{S}\right)}$ 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 ${\displaystyle \mathbb {R} ^{n}}$ with its Borel sets and ${\displaystyle \mathbb {R} ^{\infty }}$ with the cylinder σ-algebra described below.

Borel and Lebesgue σ-algebras

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 ${\displaystyle \mathbb {R} ^{n},}$ another σ-algebra is of importance: that of all Lebesgue measurable sets. This σ-algebra contains more sets than the Borel σ-algebra on ${\displaystyle \mathbb {R} ^{n}}$ and is preferred in integration theory, as it gives a complete measure space.

Product σ-algebra

Let ${\displaystyle (X_{1},\Sigma _{1})}$ and ${\displaystyle (X_{2},\Sigma _{2})}$ be two measurable spaces. The σ-algebra for the corresponding product space ${\displaystyle X_{1}\times X_{2}}$ is called the product σ-algebra and is defined by $${\displaystyle \Sigma _{1}\times \Sigma _{2}=\sigma \left(\left\{B_{1}\times B_{2}:B_{1}\in \Sigma _{1},B_{2}\in \Sigma _{2}\right\}\right).}$$

Observe that ${\displaystyle \left\{B_{1}\times B_{2}:B_{1}\in \Sigma _{1},B_{2}\in \Sigma _{2}\right\}}$ is a π-system.

The Borel σ-algebra for ${\displaystyle R^{n}}$ is generated by half-infinite rectangles and by finite rectangles. For example, $${\displaystyle {\mathcal {B}}(\mathbb {R} ^{n})=\sigma \left(\left\{(-\infty ,b_{1}]\times \cdots \times (-\infty ,b_{n}]:b_{i}\in \mathbb {R} \right\}\right)=\sigma \left(\left\{(a_{1},b_{1}]\times \cdots \times (a_{n},b_{n}]:a_{i},b_{i}\in \mathbb {R} \right\}\right).}$$

For each of these two examples, the generating family is a π-system.

σ-algebra generated by cylinder sets

Suppose $${\displaystyle X\subseteq \mathbb {R} ^{\mathbb {T} }=\{f:f{\text{ is a function from }}\mathbb {T} {\text{ to }}\mathbb {R} \}}$$ is a set of real-valued functions on ${\displaystyle \mathbb {T} }$. Let ${\displaystyle {\mathcal {B}}(\mathbb {R} )}$ denote the Borel subsets of ${\displaystyle \mathbb {R} .}$ For each ${\displaystyle \{t_{i}\}_{i=1}^{n}\subseteq \mathbb {T} }$ and ${\displaystyle \{B_{i}\}_{i=1}^{n}\subseteq {\mathcal {B}}(\mathbb {R} )}$ a cylinder subset of ${\displaystyle X}$ is a finitely restricted set defined as $${\displaystyle C_{t_{1},\dots ,t_{n}}(B_{1},\dots ,B_{n})=\{f\in X:f(t_{i})\in B_{i},1\leq i\leq n\}.}$$

For each ${\displaystyle \{t_{i}\}_{i=1}^{n}\subseteq \mathbb {T} ,}$ $${\displaystyle \{C_{t_{1},\dots ,t_{n}}(B_{1},\dots ,B_{n}):B_{i}\in {\mathcal {B}}(\mathbb {R} ),1\leq i\leq n\}}$$ is a π-system that generates a σ-algebra ${\textstyle \Sigma _{t_{1},\dots ,t_{n}}.}$ Then the family of subsets $${\displaystyle {\mathcal {F}}_{X}=\bigcup _{n=1}^{\infty }\bigcup _{t_{i}\in \mathbb {T} ,i\leq n}\Sigma _{t_{1},\dots ,t_{n}}}$$ is an algebra that generates the cylinder σ-algebra for ${\displaystyle X.}$ This σ-algebra is a subalgebra of the Borel σ-algebra determined by the product topology of ${\displaystyle \mathbb {R} ^{\mathbb {T} }}$ restricted to ${\displaystyle X.}$

An important special case is when ${\displaystyle \mathbb {T} }$ is the set of natural numbers and ${\displaystyle X}$ is a set of real-valued sequences. In this case, it suffices to consider the cylinder sets $${\displaystyle C_{n}(B_{1},\dots ,B_{n})=(B_{1}\times \cdots \times B_{n}\times \mathbb {R} ^{\infty })\cap X=\{(x_{1},x_{2},\dots ,x_{n},x_{n+1},\dots )\in X:x_{i}\in B_{i},1\leq i\leq n\},}$$ for which $${\displaystyle \Sigma _{n}=\sigma (\{C_{n}(B_{1},\dots ,B_{n}):B_{i}\in {\mathcal {B}}(\mathbb {R} ),1\leq i\leq n\})}$$ is a non-decreasing sequence of σ-algebras.

σ-algebra generated by random variable or vector

Suppose ${\displaystyle (\Omega ,\Sigma ,\mathbb {P} )}$ is a probability space. If ${\textstyle Y:\Omega \to \mathbb {R} ^{n}}$ is measurable with respect to the Borel σ-algebra on ${\displaystyle \mathbb {R} ^{n}}$ then ${\displaystyle Y}$ is called a random variable (${\displaystyle n=1}$) or random vector (${\displaystyle n>1}$). The σ-algebra generated by ${\displaystyle Y}$ is $${\displaystyle \sigma (Y)=\{Y^{-1}(A):A\in {\mathcal {B}}(\mathbb {R} ^{n})\}.}$$

σ-algebra generated by a stochastic process

Suppose ${\displaystyle (\Omega ,\Sigma ,\mathbb {P} )}$ is a probability space and ${\displaystyle \mathbb {R} ^{\mathbb {T} }}$ is the set of real-valued functions on ${\displaystyle \mathbb {T} }$. If ${\textstyle Y:\Omega \to X\subseteq \mathbb {R} ^{\mathbb {T} }}$ is measurable with respect to the cylinder σ-algebra ${\displaystyle \sigma ({\mathcal {F}}_{X})}$ (see above) for ${\displaystyle X,}$ then ${\displaystyle Y}$ is called a stochastic process or random process. The σ-algebra generated by ${\displaystyle Y}$ is $${\displaystyle \sigma (Y)=\left\{Y^{-1}(A):A\in \sigma ({\mathcal {F}}_{X})\right\}=\sigma (\{Y^{-1}(A):A\in {\mathcal {F}}_{X}\}),}$$ the σ-algebra generated by the inverse images of cylinder sets.

See also

• δ-ring – Ring closed under countable intersections
• Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
• Join (sigma algebra) – Subsets that are closed under complement and countable unionsPages displaying short descriptions of redirect targets
• 𝜆-system (Dynkin system) – Family closed under complements and countable disjoint unions
• Measurable function – Kind of mathematical function
• Monotone class – theoremPages displaying wikidata descriptions as a fallbackPages displaying short descriptions with no spaces
• π-system – Family of sets closed under intersection
• Ring of sets – Family closed under unions and relative complements
• Sample space – Set of all possible outcomes or results of a statistical trial or experiment
• 𝜎-ideal – Family closed under subsets and countable unions
• 𝜎-ring – Family of sets closed under countable unions
• 𝜎 additivity – Mapping functionPages displaying short descriptions of redirect targets

References

1. "Probability, Mathematical Statistics, Stochastic Processes". Random. University of Alabama in Huntsville, Department of Mathematical Sciences. Retrieved 30 March 2016.
2. Billingsley, Patrick (2012). Probability and Measure (Anniversary ed.). Wiley. ISBN 978-1-118-12237-2.
3. Rudin, Walter (1987). Real & Complex Analysis. McGraw-Hill. ISBN 978-0-07-054234-1.
4. Vestrup, Eric M. (2009). The Theory of Measures and Integration. John Wiley & Sons. p. 12. ISBN 978-0-470-31795-2.
5. Džamonja, Mirna; Kunen, Kenneth (1995). "Properties of the class of measure separable compact spaces" (PDF). Fundamenta Mathematicae: 262. If ${\displaystyle \mu }$ is a Borel measure on ${\displaystyle X}$, the measure algebra of ${\displaystyle (X,\mu )}$ is the Boolean algebra of all Borel sets modulo ${\displaystyle \mu }$-null sets. If ${\displaystyle \mu }$ 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 ${\displaystyle \mu }$ is separable if and only if this metric space is separable as a topological space.
6. Fischer, Tom (2013). "On simple representations of stopping times and stopping time sigma-algebras". Statistics and Probability Letters. 83 (1): 345–349. arXiv:1112.1603. doi:10.1016/j.spl.2012.09.024.
7. Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). Springer. p. 7. ISBN 978-0-387-95313-7.

• Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020.

External links

• "Algebra of sets", Encyclopedia of Mathematics, EMS Press, 2001
• Sigma Algebra at PlanetMath.

Families ${\displaystyle {\mathcal {F}}}$ of sets over ${\displaystyle \Omega }$ v t e
Is necessarily true of ${\displaystyle {\mathcal {F}}\colon }$ or, is ${\displaystyle {\mathcal {F}}}$ closed under:	Directed by ${\displaystyle \,\supseteq }$	${\displaystyle A\cap B}$	${\displaystyle A\cup B}$	${\displaystyle B\setminus A}$	${\displaystyle \Omega \setminus A}$	${\displaystyle A_{1}\cap A_{2}\cap \cdots }$	${\displaystyle A_{1}\cup A_{2}\cup \cdots }$	${\displaystyle \Omega \in {\mathcal {F}}}$	${\displaystyle \varnothing \in {\mathcal {F}}}$	F.I.P.
π-system
Semiring										Never
Semialgebra (Semifield)										Never
Monotone class						only if ${\displaystyle A_{i}\searrow }$	only if ${\displaystyle A_{i}\nearrow }$
𝜆-system (Dynkin System)				only if ${\displaystyle A\subseteq B}$			only if ${\displaystyle A_{i}\nearrow }$ or they are disjoint			Never
Ring (Order theory)
Ring (Measure theory)										Never
δ-Ring										Never
𝜎-Ring										Never
Algebra (Field)										Never
𝜎-Algebra (𝜎-Field)										Never
Dual ideal
Filter				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Prefilter (Filter base)				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Filter subbase				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Open Topology							(even arbitrary ${\displaystyle \cup }$)			Never
Closed Topology						(even arbitrary ${\displaystyle \cap }$)				Never
Is necessarily true of ${\displaystyle {\mathcal {F}}\colon }$ or, is ${\displaystyle {\mathcal {F}}}$ closed under:	directed downward	finite intersections	finite unions	relative complements	complements in ${\displaystyle \Omega }$	countable intersections	countable unions	contains ${\displaystyle \Omega }$	contains ${\displaystyle \varnothing }$	Finite Intersection Property
Additionally, a semiring is a π-system where every complement ${\displaystyle B\setminus A}$ is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F}}.}$ A semialgebra is a semiring where every complement ${\displaystyle \Omega \setminus A}$ is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F}}.}$ ${\displaystyle A,B,A_{1},A_{2},\ldots }$ are arbitrary elements of ${\displaystyle {\mathcal {F}}}$ and it is assumed that ${\displaystyle {\mathcal {F}}\neq \varnothing .}$

• Measure theory
• Experiment (probability theory)
• Set families
• Boolean algebra