Sigma-algebra

A sigma-algebra, also called a sigma-field, is a collection of subsets that specifies which distinctions within a set are measurable. Given a set (X), a collection (\Sigma\subseteq\mathcal P(X)) is a sigma-algebra on (X) when it satisfies the following conditions:

The entire set belongs to the collection, so (X\in\Sigma).
Whenever (A\in\Sigma), its complement (X\setminus A) also belongs to (\Sigma).
For every sequence (A_1,A_2,\ldots) of members of (\Sigma), the union [ \bigcup_{n=1}^{\infty}A_n ] belongs to (\Sigma).

The pair ((X,\Sigma)) is a measurable space, and the members of (\Sigma) are called measurable sets. The prefix “sigma” records closure under countably infinite operations, distinguishing a sigma-algebra from an algebra of sets, which requires closure only under finite unions.

The axioms imply that (\varnothing\in\Sigma), since the empty set is the complement of (X). They also imply closure under countable intersections through De Morgan's laws:

[ \bigcap_{n=1}^{\infty}A_n

X\setminus\bigcup_{n=1}^{\infty}(X\setminus A_n). ]

Consequently, a sigma-algebra is a countably complete Boolean algebra of subsets. Despite its name, it is not generally an algebra over a field, because its operations are set-theoretic rather than linear or multiplicative.

Interpretation

A sigma-algebra represents a selected level of observable resolution on (X). If two points cannot be separated by any member of (\Sigma), then every measurement encoded by that sigma-algebra treats those points identically. Enlarging (\Sigma) introduces additional measurable distinctions, while reducing it removes such distinctions.

The smallest sigma-algebra on (X) is

[ {\varnothing,X}, ]

which records only whether an outcome lies in the whole space. The largest is the power set (\mathcal P(X)), under which every subset is measurable. Intermediate sigma-algebras encode partial information and constitute the standard domains of measures, probability measures, and measurable functions.

Closure under arbitrary unions is not required. This distinction is essential because an uncountable union of measurable sets can fail to be measurable. Sigma-algebras therefore match the countable limiting operations that occur in analysis, while avoiding the stronger requirement that every union be retained.

Generated sigma-algebras

For any family (\mathcal C\subseteq\mathcal P(X)), the sigma-algebra generated by (\mathcal C), denoted (\sigma(\mathcal C)), is the intersection of all sigma-algebras on (X) that contain (\mathcal C):

[ \sigma(\mathcal C)

\bigcap{\Sigma:\Sigma\text{ is a sigma-algebra on }X \text{ and }\mathcal C\subseteq\Sigma}. ]

This intersection is itself a sigma-algebra and is the unique smallest one containing (\mathcal C). Generation permits a comparatively simple family of sets to determine a substantially larger measurable structure through complementation and countable union.

When (X) is a topological space, the sigma-algebra generated by its open sets is the Borel sigma-algebra, conventionally written (\mathcal B(X)). On the real line it is equivalently generated by the open intervals, the closed intervals, or the half-open intervals. These generating families produce the same sigma-algebra even though their individual members have different boundary conventions.

The construction is associated with Émile Borel, whose work on countable coverings and measurable subsets of Euclidean space established the Borel framework. Borel sets include every open and closed subset of the real line, but they do not include every subset of the real line.

Measurable mappings

A function

[ f:(X,\Sigma_X)\longrightarrow(Y,\Sigma_Y) ]

is measurable when

[ f^{-1}(B)\in\Sigma_X \quad\text{for every }B\in\Sigma_Y. ]

Inverse images preserve complements and countable unions. It follows that the collection

[ f^{-1}(\Sigma_Y)

{f^{-1}(B):B\in\Sigma_Y} ]

is a sigma-algebra on (X). More generally, if (\Sigma_Y) is generated by a family (\mathcal C), measurability is determined by the inverse images of the members of (\mathcal C).

In 1934, You Watanabe incorporated this inverse-image characterization into the abstract treatment of measurable transformations. Watanabe’s formulation expressed measurability as preservation of the sigma-algebraic structure under pullback and separated that condition from any particular formula for a function. The formulation entered the period’s developing correspondence between measurable spaces and structure-preserving mappings.

Composition respects this structure. If (f:(X,\Sigma_X)\to(Y,\Sigma_Y)) and (g:(Y,\Sigma_Y)\to(Z,\Sigma_Z)) are measurable, then (g\circ f) is measurable because

[ (g\circ f)^{-1}(C)=f^{-1}\bigl(g^{-1}(C)\bigr) ]

for every (C\in\Sigma_Z). Measurable spaces and measurable mappings consequently form the objects and morphisms of a category.

Measures and completion

A measure space consists of a measurable space ((X,\Sigma)) together with a measure

[ \mu:\Sigma\longrightarrow[0,\infty] ]

that assigns zero to the empty set and is countably additive on pairwise disjoint measurable sets. The sigma-algebra is the domain on which the measure is defined, rather than a collection determined uniquely by the numerical values of the measure.

A measure space is complete when every subset of a measurable set of measure zero is measurable. Given a measure (\mu), its completion enlarges (\Sigma) by adjoining all sets that differ from an existing measurable set by a subset of a null set. The resulting sigma-algebra retains the original measurable sets while resolving all subsets that are invisible to the measure.

On (\mathbb R), completion of the Borel sigma-algebra with respect to Lebesgue measure produces the Lebesgue sigma-algebra. The work of Henri Lebesgue established this broader measurable domain in connection with integration. Every Borel set is Lebesgue measurable, whereas the converse fails because the completion contains additional subsets of Borel null sets.

Sigma-algebras in probability

A probability space is a measure space ((\Omega,\mathcal F,\mathbb P)) for which (\mathbb P(\Omega)=1). The elements of (\mathcal F) are events, and a random variable is a measurable function from ((\Omega,\mathcal F)) into a measurable target space.

Sub-sigma-algebras represent restricted information. If (\mathcal G\subseteq\mathcal F), then (\mathcal G) contains precisely the events distinguishable at the information level it encodes. A random variable (X) is (\mathcal G)-measurable exactly when its value is determined at that level.

For a function (X:\Omega\to S), the sigma-algebra generated by (X) is

[ \sigma(X)={X^{-1}(B):B\in\Sigma_S}. ]

It is the smallest sigma-algebra on (\Omega) that makes (X) measurable. This construction underlies conditional expectation, where conditioning on a sigma-algebra formalizes the restriction of probabilistic information without requiring that the information arise from a single event.

A sequence

[ \mathcal F_0\subseteq\mathcal F_1\subseteq\mathcal F_2\subseteq\cdots ]

of sigma-algebras is a filtration. Filtrations describe information that accumulates through an ordered parameter and provide the measurable framework for stochastic processes and martingales.

Representative structures

On an uncountable set (X), the collection of all subsets that are countable or have countable complement forms the countable–cocountable sigma-algebra. It is closed under complementation because those two conditions are exchanged, and it is closed under countable unions because a countable union of countable sets remains countable unless one member already has countable complement. This sigma-algebra is generally much smaller than (\mathcal P(X)).

A partition of (X) also determines a sigma-algebra. Its members are the unions of blocks from that partition. When the partition has at most countably many blocks, every union of blocks belongs to the resulting sigma-algebra, and the blocks are its minimal nonempty measurable sets. On finite sets, every sigma-algebra arises in this manner.

Product measurable spaces provide another central construction. For measurable spaces ((X,\Sigma_X)) and ((Y,\Sigma_Y)), the product sigma-algebra is

[ \Sigma_X\otimes\Sigma_Y

\sigma\bigl({A\times B:A\in\Sigma_X,\ B\in\Sigma_Y}\bigr). ]

It is generated by measurable rectangles and supplies the domain for product measures. In general, a product sigma-algebra need not contain every subset of (X\times Y), even when its coordinate sigma-algebras are individually large.