A sigma-algebra, or σ-algebra, on a set is a collection of subsets containing the whole set and closed under complements and countable unions. It specifies which subsets are regarded as measurable, without itself assigning them sizes or probabilities. Sigma-algebras are foundational in measure theory and probability theory, where they provide domains on which measures and probabilities are defined. (math.ucdavis.edu)
Definition and closure properties
Let (X) be a set and (\mathcal F) a collection of its subsets. The collection (\mathcal F) is a sigma-algebra on (X) if:
- (X\in\mathcal F).
- Whenever (A\in\mathcal F), its complement (X\setminus A) belongs to (\mathcal F).
- Whenever (A_1,A_2,\ldots\in\mathcal F), their union satisfies [ \bigcup_{n=1}^{\infty}A_n\in\mathcal F. ]
These conditions also imply that (\varnothing\in\mathcal F), that countable intersections belong to (\mathcal F), and that (A\setminus B\in\mathcal F) whenever (A,B\in\mathcal F). Closure under intersections follows from De Morgan’s laws: [ \bigcap_{n=1}^{\infty}A_n
X\setminus\bigcup_{n=1}^{\infty}(X\setminus A_n). ] Finite unions and intersections are special cases of the countable operations. (math.ucdavis.edu)
An algebra of sets requires closure under complements and finite unions; a sigma-algebra additionally requires countable unions. Neither definition requires closure under arbitrary uncountable unions. The distinction allows measurable sets to remain closed under many operations involving sequences while excluding some subsets of the underlying space. (math.ucdavis.edu)
Examples and generated sigma-algebras
The smallest sigma-algebra on (X) is ({\varnothing,X}), called the trivial sigma-algebra. The largest is the power set (\mathcal P(X)), consisting of every subset of (X). For a single subset (A\subseteq X), the collection [ {\varnothing,A,X\setminus A,X} ] is a sigma-algebra, with duplicate elements omitted. (math.ucdavis.edu)
For any collection (\mathcal C\subseteq\mathcal P(X)), its generated sigma-algebra, denoted (\sigma(\mathcal C)), is the smallest sigma-algebra containing (\mathcal C). It exists because intersections of sigma-algebras on the same underlying set are sigma-algebras: [ \sigma(\mathcal C)= \bigcap{\mathcal F:\mathcal C\subseteq\mathcal F,\ \mathcal F\text{ is a sigma-algebra on }X}. ] Thus a generating collection specifies a measurable structure without requiring every measurable set to be listed explicitly. (math.ucdavis.edu)
For example, take (X={1,2,3,4}) and the partition ({{1,2},{3,4}}). The generated sigma-algebra contains exactly the unions of these two blocks: [ {\varnothing,{1,2},{3,4},X}. ] This follows directly from the definition: the displayed collection is already closed under complements and unions.
Borel sets and completion
In a topological space, the Borel sigma-algebra is generated by the open sets. Its members are called Borel sets. On the real line, the same sigma-algebra is generated by open intervals, or by half-lines ((-\infty,a]). It contains all open and closed sets, together with sets obtained through repeated countable unions and complements. (math.ucdavis.edu)
A measure space ((X,\mathcal F,\mu)) is complete if every subset of a measurable null set is measurable. Completion enlarges (\mathcal F) to include these subsets and extends the measure by assigning them measure zero. Completeness therefore depends on the measure, not merely on the sigma-algebra. (math.ucdavis.edu)
The completion of the Borel sigma-algebra on (\mathbb R^n) with respect to Lebesgue measure is the Lebesgue sigma-algebra. It strictly contains the Borel sigma-algebra: some subsets of Borel null sets are not Borel. Nevertheless, not every subset of (\mathbb R^n) is Lebesgue measurable. (math.ucdavis.edu)
Measurable spaces and functions
The pair ((X,\mathcal F)) is a measurable space. Unlike a measure space, it specifies measurable subsets but supplies no measure. A measurable function (f:(X,\mathcal F)\to(Y,\mathcal G)) satisfies [ f^{-1}(B)\in\mathcal F \qquad\text{for every }B\in\mathcal G. ] This is a condition on inverse images, not on images. Between topological spaces equipped with their Borel sigma-algebras, every continuous function is measurable. Measurability is also central to defining the Lebesgue integral. (math.ucdavis.edu)
For two measurable spaces, the product sigma-algebra is [ \mathcal F\otimes\mathcal G
\sigma{A\times B:A\in\mathcal F,\ B\in\mathcal G}. ] It is generated by measurable rectangles, rather than consisting only of rectangles. For Euclidean spaces, the product of their Borel sigma-algebras equals the Borel sigma-algebra of the product space. Products of complete measure spaces, however, need not be complete. (math.ucdavis.edu)
Probability and information
In a probability space ((\Omega,\mathcal F,P)), (\Omega) is the sample space, (\mathcal F) contains the events, and (P) assigns their probabilities. A random variable (Z) taking real values is measurable relative to (\mathcal F) and the Borel sigma-algebra. Its generated sigma-algebra, [ \sigma(Z)={Z^{-1}(B):B\in\mathcal B(\mathbb R)}, ] is the smallest sigma-algebra making (Z) measurable. It represents events determined by observing (Z). (math.ucdavis.edu)
A sub-sigma-algebra (\mathcal G\subseteq\mathcal F) can represent restricted information. For an integrable random variable (Z), conditional expectation (E[Z\mid\mathcal G]) is (\mathcal G)-measurable and preserves the integral of (Z) over every event in (\mathcal G). Conditioning on the trivial sigma-algebra gives the constant expected value (E[Z]). (ocw.mit.edu)
An increasing family of sigma-algebras is a filtration, describing information available over time. Filtrations supply the information structure used to define martingales and to study stochastic processes. (ocw.mit.edu)