Σ-algebra#Simple set-based examples

If $X\; =\; \backslash \{a,\; b,\; c,\; d\backslash \}$ one possible σ-algebra on $X$ is $\backslash Sigma\; =\; \backslash \{\backslash varnothing,\; \backslash \{a,\; b\backslash \},\; \backslash \{c,\; d\backslash \},\; \backslash \{a,\; b,\; c,\; d\backslash \}\backslash \},$ where $\backslash varnothing$ is the empty set. In general, a finite algebra is always a σ-algebra.

If $\backslash \{A\_1,\; A\_2,\; A\_3,\; \backslash ldots\backslash \},$ is a countable partition of $X$ 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 (a construction known as the Borel hierarchy).

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 $X$ is a function that assigns a non-negative 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 {{em|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. Non-empty collections of sets with these properties are called σ-algebras.

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 {{em|limit supremum}} or {{em|outer limit}} of a sequence $A\_1,\; A\_2,\; A\_3,\; \backslash ldots$ of subsets of $X$ is $$\backslash limsup\_\{n\backslash to\backslash infty\}\; A\_n\; =\; \backslash bigcap\_\{n=1\}^\backslash infty\; \backslash bigcup\_\{m=n\}^\backslash infty\; A\_m\; =\; \backslash bigcap\_\{n=1\}^\backslash infty\; A\_n\; \backslash cup\; A\_\{n+1\}\; \backslash cup\; \backslash cdots.$$ It consists of all points $x$ that are in infinitely many of these sets (or equivalently, that are in Cofinal (mathematics) of them). That is, $x\; \backslash in\; \backslash limsup\_\{n\backslash to\backslash infty\}\; A\_n$ if and only if there exists an infinite subsequence $A\_\{n\_1\},\; A\_\{n\_2\},\; \backslash ldots$ (where ) of sets that all contain $x;$ that is, such that $x\; \backslash in\; A\_\{n\_1\}\; \backslash cap\; A\_\{n\_2\}\; \backslash cap\; \backslash cdots.$
• The {{em|limit infimum}} or {{em|inner limit}} of a sequence $A\_1,\; A\_2,\; A\_3,\; \backslash ldots$ of subsets of $X$ is $$\backslash liminf\_\{n\backslash to\backslash infty\}\; A\_n\; =\; \backslash bigcup\_\{n=1\}^\backslash infty\; \backslash bigcap\_\{m=n\}^\backslash infty\; A\_m\; =\; \backslash bigcup\_\{n=1\}^\backslash infty\; A\_n\; \backslash cap\; A\_\{n+1\}\; \backslash cap\; \backslash cdots.$$ It consists of all points that are in all but finitely many of these sets (or equivalently, that are {{em|eventually}} in all of them). That is, $x\; \backslash in\; \backslash liminf\_\{n\backslash to\backslash infty\}\; A\_n$ if and only if there exists an index $N\; \backslash in\; \backslash N$ such that $A\_N,\; A\_\{N+1\},\; \backslash ldots$ all contain $x;$ that is, such that $x\; \backslash in\; A\_N\; \backslash cap\; A\_\{N+1\}\; \backslash cap\; \backslash cdots.$

The inner limit is always a subset of the outer limit: $$\backslash liminf\_\{n\backslash to\backslash infty\}\; A\_n\; ~\backslash subseteq~\; \backslash limsup\_\{n\backslash to\backslash infty\}\; A\_n.$$

If these two sets are equal then their limit $\backslash lim\_\{n\backslash to\backslash infty\}\; A\_n$ exists and is equal to this common set:

$$\backslash lim\_\{n\backslash to\backslash infty\}\; A\_n\; :=\; \backslash liminf\_\{n\backslash to\backslash infty\}\; A\_n\; =\; \backslash limsup\_\{n\backslash to\backslash infty\}\; A\_n.$$

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 $\backslash Sigma,\; \backslash Sigma\text{'}$ are σ-algebras on $X$, then $\backslash Sigma\text{'}$ is a sub σ-algebra of $\backslash Sigma$ if $\backslash Sigma\text{'}\; \backslash subseteq\; \backslash Sigma$.

The Bernoulli process provides a simple example. This consists of a sequence of random coin flips, coming up Heads ($H$) or Tails ($T$), of unbounded length. The sample space Ω consists of all possible infinite sequences of $H$ or $T:$

$$\backslash Omega\; =\; \backslash \{H,\; T\backslash \}^\backslash infty\; =\; \backslash \{(x\_1,\; x\_2,\; x\_3,\; \backslash dots)\; :\; x\_i\; \backslash in\; \backslash \{H,\; T\backslash \},\; i\; \backslash geq\; 1\backslash \}.$$

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 $n$ coin flips. This sequence of subalgebras is given by

\mathcal{G}_n = \{A \times \{ \Omega \} : A \subseteq \{H, T\}^n\}

Each of these is finer than the last, and so can be ordered as a filtration

$$\backslash mathcal\{G\}\_0\; \backslash subseteq\; \backslash mathcal\{G\}\_1\; \backslash subseteq\; \backslash mathcal\{G\}\_2\; \backslash subseteq\; \backslash cdots\; \backslash subseteq\; \backslash mathcal\{G\}\_\backslash infty$$

The first subalgebra $\backslash mathcal\{G\}\_0\; =\; \backslash \{\backslash varnothing,\; \backslash Omega\backslash \}$ is the trivial algebra: it has only two elements in it, the empty set and the total space. The second subalgebra $\backslash mathcal\{G\}\_1$ has four elements: the two in $\backslash mathcal\{G\}\_0$ plus two more: sequences that start with $H$ and sequences that start with $T$. Each subalgebra is finer than the last. The $n$'th subalgebra contains $2^\{n+1\}$ elements: it divides the total space $\backslash Omega$ into all of the possible sequences that might have been observed after $n$ flips, including the possible non-observation of some of the flips.

The limiting algebra $\backslash mathcal\{G\}\_\backslash infty$ 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 $\backslash \{H,T\backslash \}^\backslash infty.$

Let $X$ be some set, and let $P(X)$ represent its power set, the set of all subsets of $X$. Then a subset $\backslash Sigma\; \backslash subseteq\; P(X)$ is called a σ-algebra if and only if it satisfies the following three properties:{{cite book | last=Rudin | first=Walter | authorlink=Walter Rudin | title=Real & Complex Analysis | publisher=McGraw-Hill | year=1987 | isbn=0-07-054234-1}}

1. $X$ is in $\backslash Sigma$.
2. $\backslash Sigma$ is closed under complementation: If some set $A$ is in $\backslash Sigma,$ then so is its complement, $X\; \backslash setminus\; A.$
3. $\backslash Sigma$ is closed under countable unions: If $A\_1,\; A\_2,\; A\_3,\; \backslash ldots$ are in $\backslash Sigma,$ then so is $A\; =\; A\_1\; \backslash cup\; A\_2\; \backslash cup\; A\_3\; \backslash cup\; \backslash cdots.$

From these properties, it follows that the σ-algebra is also closed under countable intersections (by applying De Morgan's laws).

It also follows that the empty set $\backslash varnothing$ is in $\backslash Sigma,$ since by (1) $X$ is in $\backslash Sigma$ and (2) asserts that its complement, the empty set, is also in $\backslash Sigma.$ Moreover, since $\backslash \{X,\; \backslash varnothing\backslash \}$ satisfies all 3 conditions, it follows that $\backslash \{X,\; \backslash varnothing\backslash \}$ is the smallest possible σ-algebra on $X.$ The largest possible σ-algebra on $X$ is $P(X).$

Elements of the σ-algebra are called measurable sets. An ordered pair $(X,\; \backslash Sigma),$ where $X$ is a set and $\backslash Sigma$ is a σ-algebra 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 σ-algebra to $[0,\; \backslash infty].$

A σ-algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin's theorem (see below).

{{See also|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 $P$ is a collection of subsets of $X$ that is closed under finitely many intersections, and
• A Dynkin system (or λ-system) $D$ is a collection of subsets of $X$ that contains $X$ and is closed under complement and under countable unions of disjoint subsets.

Dynkin's π-λ theorem says, if $P$ is a π-system and $D$ is a Dynkin system that contains $P,$ then the σ-algebra $\backslash sigma(P)$ generated by $P$ is contained in $D.$ Since certain π-systems 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 π-λ Theorem then implies that all sets in $\backslash sigma(P)$ enjoy the property, avoiding the task of checking it for an arbitrary set in $\backslash sigma(P).$

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 $X$ with the Lebesgue-Stieltjes integral typically associated with computing the probability:

$$\backslash mathbb\{P\}(X\backslash in\; A)\; =\; \backslash int\_A\; \backslash ,F(dx)$$ for all $A$ in the Borel σ-algebra on $\backslash Reals,$

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

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

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:

$$\backslash bigwedge\_\{\backslash alpha\; \backslash in\; \backslash mathcal\{A\}\}\; \backslash Sigma\_\backslash alpha.$$

Sketch of Proof: Let $\backslash Sigma^*$ denote the intersection. Since $X$ is in every $\backslash Sigma\_\backslash alpha,\; \backslash Sigma^*$ is not empty. Closure under complement and countable unions for every $\backslash Sigma\_\backslash alpha$ implies the same must be true for $\backslash Sigma^*.$ Therefore, $\backslash Sigma^*$ 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

$$\backslash bigvee\_\{\backslash alpha\; \backslash in\; \backslash mathcal\{A\}\}\; \backslash Sigma\_\backslash alpha\; =\; \backslash sigma\backslash left(\backslash bigcup\_\{\backslash alpha\; \backslash in\; \backslash mathcal\{A\}\}\; \backslash Sigma\_\backslash alpha\backslash right).$$

A π-system that generates the join is

$$\backslash mathcal\{P\}\; =\; \backslash left\backslash \{\backslash bigcap\_\{i=1\}^n\; A\_i\; :\; A\_i\; \backslash in\; \backslash Sigma\_\{\backslash alpha\_i\},\; \backslash alpha\_i\; \backslash in\; \backslash mathcal\{A\},\backslash \; n\; \backslash geq\; 1\backslash right\backslash \}.$$

Sketch of Proof: By the case $n\; =\; 1,$ it is seen that each $\backslash Sigma\_\backslash alpha\backslash subset\backslash mathcal\{P\},$ so

$$\backslash bigcup\_\{\backslash alpha\; \backslash in\; \backslash mathcal\{A\}\}\; \backslash Sigma\_\backslash alpha\; \backslash subseteq\; \backslash mathcal\{P\}.$$

This implies

$$\backslash sigma\backslash left(\backslash bigcup\_\{\backslash alpha\; \backslash in\; \backslash mathcal\{A\}\}\; \backslash Sigma\_\backslash alpha\backslash right)\; \backslash subseteq\; \backslash sigma(\backslash mathcal\{P\})$$

by the definition of a σ-algebra generated by a collection of subsets. On the other hand,

$$\backslash mathcal\{P\}\; \backslash subseteq\; \backslash sigma\backslash left(\backslash bigcup\_\{\backslash alpha\; \backslash in\; \backslash mathcal\{A\}\}\; \backslash Sigma\_\backslash alpha\backslash right)$$

which, by Dynkin's π-λ theorem, implies

$$\backslash sigma(\backslash mathcal\{P\})\; \backslash subseteq\; \backslash sigma\backslash left(\backslash bigcup\_\{\backslash alpha\; \backslash in\; \backslash mathcal\{A\}\}\; \backslash Sigma\_\backslash alpha\backslash right).$$

Suppose $Y$ is a subset of $X$ and let $(X,\; \backslash Sigma)$ be a measurable space.

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

A σ-algebra $\backslash Sigma$ is just a σ-ring that contains the universal set $X.${{cite book| last=Vestrup |first=Eric M. |title=The Theory of Measures and Integration |year=2009 |publisher=John Wiley & Sons |isbn=978-0-470-31795-2 |page=12}} 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.

σ-algebras are sometimes denoted using calligraphic capital letters, or the Fraktur typeface. Thus $(X,\; \backslash Sigma)$ may be denoted as $\backslash scriptstyle(X,\backslash ,\backslash mathcal\{F\})$ or $\backslash scriptstyle(X,\backslash ,\backslash mathfrak\{F\}).$

A separable $\backslash sigma$-algebra (or separable $\backslash sigma$-field) is a $\backslash sigma$-algebra $\backslash mathcal\{F\}$ that is a separable space when considered as a metric space with metric $\backslash rho(A,B)\; =\; \backslash mu(A\; \backslash mathbin\{\backslash triangle\}\; B)$ for $A,B\; \backslash in\; \backslash mathcal\{F\}$ and a given finite measure $\backslash mu$ (and with $\backslash triangle$ being the symmetric difference operator).{{cite journal|last1=Džamonja|first1=Mirna|last2=Kunen|first2=Kenneth|authorlink2=Kenneth Kunen|title=Properties of the class of measure separable compact spaces| journal=Fundamenta Mathematicae|year=1995|pages=262|url=https://archive.uea.ac.uk/~h020/fundamenta.pdf|quote=If $\backslash mu$ is a Borel measure on $X,$ the measure algebra of $(X,\backslash mu)$ is the Boolean algebra of all Borel sets modulo $\backslash mu$-null sets. If $\backslash mu$ 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 $\backslash mu$ is separable if and only if this metric space is separable as a topological space.}} Any $\backslash sigma$-algebra generated by a countable collection of sets is separable, but the converse need not hold. For example, the Lebesgue $\backslash sigma$-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. 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.

Let $X$ be any set.

• The family consisting only of the empty set and the set $X,$ called the minimal or trivial σ-algebra over $X.$
• The power set of $X,$ called the discrete σ-algebra.
• The collection $\backslash \{\backslash varnothing,\; A,\; X\; \backslash setminus\; A,\; X\backslash \}$ is a simple σ-algebra generated by the subset $A.$
• The collection of subsets of $X$ which are countable or whose complements are countable is a σ-algebra (which is distinct from the power set of $X$ if and only if $X$ is uncountable). This is the σ-algebra 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 σ-algebra.

A stopping time $\backslash tau$ can define a $\backslash sigma$-algebra $\backslash mathcal\{F\}\_\backslash tau,$ the

so-called stopping time sigma-algebra, which in a filtered probability space describes the information up to the random time $\backslash tau$ 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 $\backslash tau$ is $\backslash mathcal\{F\}\_\{\backslash tau\}.${{cite journal|last=Fischer|first=Tom|title=On simple representations of stopping times and stopping time sigma-algebras|journal=Statistics and Probability Letters|year=2013|volume=83|issue=1|pages=345–349|doi=10.1016/j.spl.2012.09.024|arxiv=1112.1603 |url=https://dx.doi.org/10.1016/j.spl.2012.09.024}}

{{anchor}}

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

If $F$ is empty, then $\backslash sigma(\backslash varnothing)\; =\; \backslash \{\backslash varnothing,\; X\backslash \}.$ Otherwise $\backslash sigma(F)$ consists of all the subsets of $X$ that can be made from elements of $F$ by a countable number of complement, union and intersection operations.

For a simple example, consider the set $X\; =\; \backslash \{1,\; 2,\; 3\backslash \}.$ Then the σ-algebra generated by the single subset $\backslash \{1\backslash \}$ is

$\backslash sigma(\backslash \{1\backslash \})\; =\; \backslash \{\backslash varnothing,\; \backslash \{1\backslash \},\; \backslash \{2,\; 3\backslash \},\; \backslash \{1,\; 2,\; 3\backslash \}\backslash \}.$

By an abuse of notation, when a collection of subsets contains only one element, $A,$ $\backslash sigma(A)$ may be written instead of $\backslash sigma(\backslash \{A\backslash \});$ in the prior example $\backslash sigma(\backslash \{1\backslash \})$ instead of $\backslash sigma(\backslash \{\backslash \{1\backslash \}\backslash \}).$ Indeed, using $\backslash sigma\backslash left(A\_1,\; A\_2,\; \backslash ldots\backslash right)$ to mean $\backslash sigma\backslash left(\backslash left\backslash \{A\_1,\; A\_2,\; \backslash ldots\backslash right\backslash \}\backslash right)$ is also quite common.

There are many families of subsets that generate useful σ-algebras. Some of these are presented here.

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

$$\backslash sigma\; (f)\; =\; \backslash left\backslash \{f^\{-1\}(S)\; \backslash ,\; :\; \backslash ,\; S\; \backslash in\; B\backslash right\backslash \}.$$

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

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 $\backslash Reals^n$ then $\backslash sigma(f)$ is generated by the family of subsets which are inverse images of intervals/rectangles in $\backslash Reals^n:$

$$\backslash sigma(f)\; =\; \backslash sigma\backslash left(\backslash left\backslash \{f^\{-1\}(\backslash left[a\_1,\; b\_1\backslash right]\; \backslash times\; \backslash cdots\; \backslash times\; \backslash left[a\_n,\; b\_n\backslash right])\; :\; a\_i,\; b\_i\; \backslash in\; \backslash Reals\backslash right\backslash \}\backslash right).$$

A useful property is the following. Assume $f$ is a measurable map from $\backslash left(X,\; \backslash Sigma\_X\backslash right)$ to $\backslash left(S,\; \backslash Sigma\_S\backslash right)$ and $g$ is a measurable map from $\backslash left(X,\; \backslash Sigma\_X\backslash right)$ to $\backslash left(T,\; \backslash Sigma\_T\backslash right).$ If there exists a measurable map $h$ from $\backslash left(T,\; \backslash Sigma\_T\backslash right)$ to $\backslash left(S,\; \backslash Sigma\_S\backslash right)$ such that $f(x)\; =\; h(g(x))$ for all $x,$ then $\backslash sigma(f)\; \backslash subseteq\; \backslash sigma(g).$ If $S$ is finite or countably infinite or, more generally, $\backslash left(S,\; \backslash Sigma\_S\backslash right)$ is a standard Borel space (for example, a separable complete metric space with its associated Borel sets), then the converse is also true.{{cite book | last=Kallenberg | first=Olav | authorlink=Olav Kallenberg | title=Foundations of Modern Probability |edition=2nd | publisher=Springer | year=2001 | isbn=0-387-95313-2|page=7}} Examples of standard Borel spaces include $\backslash Reals^n$ with its Borel sets and $\backslash Reals^\backslash infty$ with the cylinder σ-algebra described below.

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

Let $\backslash left(X\_1,\; \backslash Sigma\_1\backslash right)$ and $\backslash left(X\_2,\; \backslash Sigma\_2\backslash right)$ be two measurable spaces. The σ-algebra for the corresponding product space $X\_1\; \backslash times\; X\_2$ is called the product σ-algebra and is defined by

$$\backslash Sigma\_1\; \backslash times\; \backslash Sigma\_2\; =\; \backslash sigma\backslash left(\backslash left\backslash \{B\_1\; \backslash times\; B\_2\; :\; B\_1\; \backslash in\; \backslash Sigma\_1,\; B\_2\; \backslash in\; \backslash Sigma\_2\backslash right\backslash \}\backslash right).$$

Observe that $\backslash \{B\_1\; \backslash times\; B\_2\; :\; B\_1\; \backslash in\; \backslash Sigma\_1,\; B\_2\; \backslash in\; \backslash Sigma\_2\backslash \}$ is a π-system.

The Borel σ-algebra for $\backslash Reals^n$ is generated by half-infinite rectangles and by finite rectangles. For example,

$$\backslash mathcal\{B\}(\backslash Reals^n)\; =\; \backslash sigma\; \backslash left(\backslash left\backslash \{(-\backslash infty,\; b\_1]\; \backslash times\; \backslash cdots\; \backslash times\; (-\backslash infty,\; b\_n]\; :\; b\_i\; \backslash in\; \backslash Reals\backslash right\backslash \}\backslash right)\; =\; \backslash sigma\backslash left(\backslash left\backslash \{\backslash left(a\_1,\; b\_1\backslash right]\; \backslash times\; \backslash cdots\; \backslash times\; \backslash left(a\_n,\; b\_n\backslash right]\; :\; a\_i,\; b\_i\; \backslash in\; \backslash Reals\backslash right\backslash \}\backslash right).$$

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

{{main|Cylinder σ-algebra}}

Suppose

$$X\; \backslash subseteq\; \backslash Reals^\{\backslash mathbb\{T\}\}\; =\; \backslash \{f\; :\; f(t)\; \backslash in\; \backslash Reals,\backslash \; t\; \backslash in\; \backslash mathbb\{T\}\backslash \}$$

is a set of real-valued functions. Let $\backslash mathcal\{B\}(\backslash Reals)$ denote the Borel subsets of $\backslash Reals.$ A cylinder subset of $X$ is a finitely restricted set defined as

$$C\_\{t\_1,\backslash dots,t\_n\}(B\_1,\backslash dots,B\_n)\; =\; \backslash left\backslash \{f\; \backslash in\; X\; :\; f(t\_i)\; \backslash in\; B\_i,\; 1\; \backslash leq\; i\; \backslash leq\; n\backslash right\backslash \}.$$

Each

$$\backslash left\backslash \{C\_\{t\_1,\backslash dots,t\_n\}\backslash left(B\_1,\; \backslash dots,\; B\_n\backslash right)\; :\; B\_i\; \backslash in\; \backslash mathcal\{B\}(\backslash Reals),\; 1\; \backslash leq\; i\; \backslash leq\; n\backslash right\backslash \}$$

is a π-system that generates a σ-algebra $\backslash textstyle\backslash Sigma\_\{t\_1,\; \backslash dots,\; t\_n\}.$ Then the family of subsets

$$\backslash mathcal\{F\}\_X\; =\; \backslash bigcup\_\{n=1\}^\backslash infty\; \backslash bigcup\_\{t\_i\; \backslash in\; \backslash mathbb\{T\},i\; \backslash leq\; n\}\backslash Sigma\_\{t\_1,\; \backslash dots,\; t\_n\}$$

is an algebra that generates the cylinder σ-algebra for $X.$ This σ-algebra is a subalgebra of the Borel σ-algebra determined by the product topology of $\backslash Reals^\{\backslash mathbb\{T\}\}$ restricted to $X.$

An important special case is when $\backslash mathbb\{T\}$ is the set of natural numbers and $X$ is a set of real-valued sequences. In this case, it suffices to consider the cylinder sets

$$C\_n\backslash left(B\_1,\; \backslash dots,\; B\_n\backslash right)\; =\; \backslash left(B\_1\; \backslash times\; \backslash cdots\; \backslash times\; B\_n\; \backslash times\; \backslash Reals^\backslash infty\backslash right)\; \backslash cap\; X\; =\; \backslash left\backslash \{\backslash left(x\_1,\; x\_2,\; \backslash ldots,\; x\_n,x\_\{n+1\},\backslash ldots\backslash right)\; \backslash in\; X\; :\; x\_i\; \backslash in\; B\_i\; ,1\; \backslash leq\; i\; \backslash leq\; n\backslash right\backslash \},$$

for which

$$\backslash Sigma\_n\; =\; \backslash sigma\backslash left(\backslash \{C\_n\backslash left(B\_1,\; \backslash dots,\; B\_n\backslash right)\; :\; B\_i\; \backslash in\; \backslash mathcal\{B\}(\backslash Reals),\; 1\; \backslash leq\; i\; \backslash leq\; n\backslash \}\backslash right)$$

is a non-decreasing sequence of σ-algebras.

The ball σ-algebra is the smallest σ-algebra containing all the open (and/or closed) balls. This is never larger than the Borel σ-algebra. Note that the two σ-algebra are equal for separable spaces. For some nonseparable spaces, some maps are ball measurable even though they are not Borel measurable, making use of the ball σ-algebra useful in the analysis of such maps.van der Vaart, A. W., & Wellner, J. A. (1996). Weak Convergence and Empirical Processes. In Springer Series in Statistics. Springer New York. https://doi.org/10.1007/978-1-4757-2545-2

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

$$\backslash sigma(Y)\; =\; \backslash left\backslash \{Y^\{-1\}(A)\; :\; A\; \backslash in\; \backslash mathcal\{B\}\backslash left(\backslash Reals^n\backslash right)\backslash right\backslash \}.$$

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

$$\backslash sigma(Y)\; =\; \backslash left\backslash \{Y^\{-1\}(A)\; :\; A\; \backslash in\; \backslash sigma\backslash left(\backslash mathcal\{F\}\_X\backslash right)\backslash right\backslash \}\; =\; \backslash sigma\backslash left(\backslash left\backslash \{Y^\{-1\}(A)\; :\; A\; \backslash in\; \backslash mathcal\{F\}\_X\backslash right\backslash \}\backslash right),$$

the σ-algebra generated by the inverse images of cylinder sets.

• {{annotated link|Measurable function}}
• {{annotated link|Sample space}}
• {{annotated link|Sigma-additive set function}}
• {{annotated link|Sigma-ring}}

{{Families of sets}}

{{Reflist}}

• {{springer|title=Algebra of sets|id=p/a011400}}
• Sigma Algebra from PlanetMath.

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Category:Boolean algebra

Category:Experiment (probability theory)

Category:Families of sets

Category:Measure theory