Σ-algebra (σ-field)

A σ-algebra, also called a σ-field or Boolean σ-algebra, on a set $X$ is a collection $\mathcal{F}$ of subsets of $X$ that contains $X$ itself and is closed under complementation and countable unions. It is the domain on which measures, including probability measures, are defined: a measure assigns sizes to exactly the sets in a σ-algebra, and the closure properties guarantee that the set operations used in measuring and in probability carry no one outside the domain.^{1 • 2}

Definition and immediate consequences

Fix a set $X$. A family $\mathcal{F} \subset \mathcal{P}(X)$ is a σ-algebra when:

1. $X \in \mathcal{F}$;
2. if $B \in \mathcal{F}$ then $X \setminus B \in \mathcal{F}$;
3. for any sequence $B_1, B_2, B_3, \dots$ of sets in $\mathcal{F}$, the union $\bigcup_{n=1}^{\infty} B_n$ belongs to $\mathcal{F}$.¹

Some texts state the axioms with countable union and set difference instead of complementation; the two formulations are equivalent.⁵ From the axioms, $\emptyset = X \setminus X \in \mathcal{F}$, and De Morgan's identity

$$\bigcap_{n=1}^{\infty} B_n = X \setminus \left( \bigcup_{n=1}^{\infty} X \setminus B_n \right)$$

gives closure under countable intersections. Closure under finite unions follows by taking $B_{n+i} = \emptyset$ for all $i$, and closure under set differences $B \setminus C$ follows from complementation and intersection.^{1 • 3} A σ-algebra is thus an algebra of sets (closed under finite operations) that additionally admits countable unions; equivalently, an algebra of sets that contains the whole space and is closed under countable union.⁵

The pair $(X, \mathcal{F})$ is called a measurable space.¹

Examples

Trivial and full σ-algebras. For any set $X$, the collection $\{\emptyset, X\}$ is a σ-algebra, called the trivial σ-algebra; it carries no information about which point of $X$ was chosen. The power set $\mathcal{P}(X)$ is also a σ-algebra, the largest possible one.² When $\Omega$ is countable, probability is typically built on the full power set: if $p \colon \Omega \to [0,1]$ with $\sum_{\omega \in \Omega} p(\omega) = 1$, then $\mathbb{P}(A) = \sum_{a \in A} p(a)$ defines a probability measure on $(\Omega, \mathcal{P}(\Omega))$.⁶

Generated σ-algebras. For any collection $\mathcal{G} \subset \mathcal{P}(X)$, the intersection of all σ-algebras containing $\mathcal{G}$ is itself a σ-algebra, written $\sigma(\mathcal{G})$ and called the σ-algebra generated by $\mathcal{G}$. It is the smallest σ-algebra containing $\mathcal{G}$, and any σ-algebra that contains $\mathcal{G}$ automatically contains $\sigma(\mathcal{G})$.¹ Two counting examples: if $X$ is partitioned into $N$ sets $X_1, \dots, X_N$, then $\sigma(\{X_1, \dots, X_N\})$ consists of all unions of the parts; and if $X$ is countable and $\mathcal{F}$ is the family of singletons, then $\sigma(\mathcal{F}) = 2^X$.⁴

The Borel σ-algebra. When $X$ is a topological space, the σ-algebra generated by the open sets is the Borel σ-algebra, denoted $\mathcal{B}(X)$; its members are the Borel sets. On $\mathbb{R}$ the open intervals alone generate it, since every open subset of $\mathbb{R}$ is a countable union of open intervals. The Borel σ-algebra contains the closed sets, countable unions of closed sets, countable intersections of open sets, and so on; no concrete description of an arbitrary Borel set is feasible, but as a rule of thumb a set that can be written down by hand is Borel. There are exceptions: a set defined using continued-fraction digits, in an example due to Luzin, is not Borel.¹ There are also subsets of $(0,1]$ that are not Borel sets, though they cannot be described explicitly.³

Why countable unions, and why not all subsets

The choice of countable rather than finite closure is what separates a σ-algebra from a plain algebra of sets. A field of sets closed only under finite unions suffices for finite probability calculations, but events of interest in analysis, such as the set of outcomes for which a sequence of averages converges, are built from countably many set operations, so the domain must be closed under them.^{3 • 7} Countable closure is also exactly what a countably additive measure needs: because a σ-algebra is closed under countable unions, the expression $\mu\left( \bigcup_n B_n \right) = \sum_n \mu(B_n)$ for disjoint $B_n$ is meaningful on its domain.⁸

Arbitrary (uncountable) unions are not required, and allowing them would be destructive.

The power set must in fact be given up on uncountable spaces. For $\Omega = [0,1]$, there is no probability measure on $\mathcal{P}(\Omega)$ with $\mathbb{P}([a,b]) = b - a$; a suitable σ-algebra $\mathcal{L}$ carrying the Lebesgue measure does support such a measure.⁶ More generally, for uncountable sample spaces there exist subsets that cannot be assigned a probability consistently; the construction of such non-measurable sets underlies the Banach–Tarski paradox.² Restricting to a σ-algebra is therefore the device that keeps probability and measure well defined.⁹

Measures and uniqueness on generated σ-algebras

A measure on a σ-algebra $\mathcal{B}$ is a map $\mu \colon \mathcal{B} \to [0,\infty]$ with $\mu(\emptyset) = 0$ that is countably additive over disjoint sequences; a measure space is a triple $(X, \mathcal{B}, \mu)$, and a probability space is the case $\mu(X) = 1$.^{8 • 4} Interesting measures are hard to construct directly because membership in a σ-algebra is hard to describe; the Lebesgue measure on the real line takes most of a chapter to build, while counting measure and point measures admit explicit formulas on the full power set.⁸

Values on a generating class determine the measure on the whole σ-algebra. If two probability measures agree on an algebra $\mathcal{A}$, they agree on $\sigma(\mathcal{A})$; this is why Lebesgue measure on $[0,1]$ is unique, since any measure agreeing with length on intervals is fixed on the algebra they generate, and that algebra generates the Borel σ-algebra.¹⁰ The standard proof tool is the monotone class theorem: the smallest monotone class containing an algebra (a family closed under monotone increasing unions and decreasing intersections) is exactly the smallest σ-algebra containing it. This lets one prove a property first on simple sets such as intervals or cylinder sets and then extend it to the generated σ-algebra.^{11 • 10}

σ-algebras as information

In probability theory a sub-σ-algebra $\mathcal{G} \subset \mathcal{F}$ is read as a stock of partial information: having observed $\mathcal{G}$ means knowing, for every $A \in \mathcal{G}$, whether $A$ occurred. The conditional expectation $\mathbb{E}[X \mid \mathcal{G}]$ is the best $\mathcal{G}$-measurable prediction of an integrable random variable $X$, characterized by $\int_A Y \, d\mathbb{P} = \int_A X \, d\mathbb{P}$ for every $A \in \mathcal{G}$ and unique up to almost-sure equality.¹² When $\mathcal{G}$ is generated by a random variable $Y$, conditioning on $\mathcal{G}$ recovers the elementary definition of $\mathbb{E}[X \mid Y]$ as a function of $Y$.¹³

A filtration is an increasing sequence of σ-algebras $\mathcal{F}_0 \subseteq \mathcal{F}_1 \subseteq \cdots$, with $\mathcal{F}_t$ encoding everything known by time $t$. In a coin-tossing market model, $\mathcal{F}_t$ is the smallest σ-algebra containing the events determined by the first $t$ tosses; the share price $S_t$ is $\mathcal{F}_t$-measurable because it depends only on those tosses, while the maximum of the price over $[0,T]$ is not $\mathcal{F}_t$-measurable for any $t < T$. A stopping time $T$ satisfies $\{T \le t\} \in \mathcal{F}_t$ for every $t$.¹³

Historical origin

The σ-algebra concept emerged from the measure theory built around 1900. In 1898 Émile Borel extended the length of an interval to a countably additive measure on a broad class of subsets of the real line, building recursively by adding complements and countable disjoint unions; the resulting family, closed under complementation and countable union, is what is now called a σ-algebra (or family of Borel). In his 1902 doctoral thesis Intégrale, longueur, aire, Henri-Léon Lebesgue simplified and extended Borel's construction, defined measurable sets by comparing inner and outer measure, showed that the measurable sets form a σ-algebra containing the Borel sets, and developed the modern integral on that foundation.¹⁴ The Borel hierarchy of definable sets, obtained from open sets by iterated countable union and complementation, was studied systematically by Lebesgue in a 1905 memoir and became the starting point of descriptive set theory.¹⁵

References

1. σ-algebras, Donald Robertson, University of Manchester lecture notes
2. 1 Axioms of probability, Probability 1 lecture notes, Clare Wallace, Durham University
3. Unit I: Measures and Sigma-Fields, Virginia Tech 6105 notes
4. Probability and Measure lecture notes, DPMMS, University of Cambridge
5. Measure and probability: Part I, N. Sanyal, University of Texas at El Paso
6. Probability spaces and Random Variables, Gautam Iyer, Carnegie Mellon University
7. Sigma-Algebras & Measures, formalCalculus
8. Measures, Donald Robertson, University of Manchester lecture notes
9. Interpretive Implications of the Sample Space, PhilSci Archive
10. Measure Theory notes, Steve Lalley, University of Chicago
11. Measure and Integration lecture notes, University of Toronto (MAT1000 notes by A. Burchard)
12. Advanced Probability lecture notes, Mathias Beiglböck, University of Vienna
13. Chapter 4: Filtrations, Conditional Expectation, and Martingales, Mathematical Finance notes, Clare Wallace, Durham University
14. La seconda rivoluzione scientifica: matematica e logica. La teoria della misura, Enciclopedia Italiana Treccani
15. The Early Development of Set Theory, Stanford Encyclopedia of Philosophy

Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Measure theory

Initially written Sep 17, 2026 · Reviewed: Sep 18, 2026; Sep 30, 2026; Oct 15, 2026 · Edited: Sep 18, 2026; Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026

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