# 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](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/sigma_algebras.html) • [2](https://www.maths.dur.ac.uk/users/clare.wallace/Prob1/01-axioms.html)
| Key fact | Detail |
|---|---|
| Definition | \( X \in \mathcal{F} \); \( B \in \mathcal{F} \) implies \( X \setminus B \in \mathcal{F} \); and \( B_1 \cup B_2 \cup \cdots \in \mathcal{F} \) for every sequence of sets in \( \mathcal{F} \)[1](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/sigma_algebras.html) |
| Consequences | Also closed under countable intersections, finite unions, finite intersections, and set differences; \( \emptyset \in \mathcal{F} \)[1](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/sigma_algebras.html) • [3](https://personal.math.vt.edu/daymv/class_homepages/6105Notes.pdf) |
| Extremes | The trivial σ-algebra \( \{\emptyset, X\} \) is the smallest; the power set \( 2^X \) is the largest[2](https://www.maths.dur.ac.uk/users/clare.wallace/Prob1/01-axioms.html) |
| Generated σ-algebra | \( \sigma(\mathcal{G}) \) is the smallest σ-algebra containing a collection \( \mathcal{G} \); every σ-algebra containing \( \mathcal{G} \) contains \( \sigma(\mathcal{G}) \)[1](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/sigma_algebras.html) |
| Borel σ-algebra | The σ-algebra generated by the open sets of a topological space; on \( \mathbb{R} \) it is also generated by the open intervals alone[1](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/sigma_algebras.html) |
| Role in probability | A probability space is a triple \( (\Omega, \mathcal{F}, \mathbb{P}) \) with \( \mathbb{P} \) a measure on \( \mathcal{F} \) and \( \mathbb{P}(\Omega) = 1 \)[4](https://www.dpmms.cam.ac.uk/study/II/Probability%2BMeasure/2019-2020/lecture-notes-PM-1-21.pdf) |
| Why not all subsets | For uncountable \( \Omega \) such as \( [0,1] \), some subsets cannot be assigned a probability consistently, so a smaller σ-algebra is used[2](https://www.maths.dur.ac.uk/users/clare.wallace/Prob1/01-axioms.html) |
## 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](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/sigma_algebras.html)
Some texts state the axioms with countable union and set difference instead of complementation; the two formulations are equivalent.[5](https://www.math.utep.edu/faculty/nsanyal/blog/measure_and_prob_1.html) 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](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/sigma_algebras.html) • [3](https://personal.math.vt.edu/daymv/class_homepages/6105Notes.pdf) 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](https://www.math.utep.edu/faculty/nsanyal/blog/measure_and_prob_1.html)
The pair \( (X, \mathcal{F}) \) is called a **measurable space**.[1](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/sigma_algebras.html)
## 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](https://www.maths.dur.ac.uk/users/clare.wallace/Prob1/01-axioms.html) 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](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html)
**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](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/sigma_algebras.html) 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](https://www.dpmms.cam.ac.uk/study/II/Probability%2BMeasure/2019-2020/lecture-notes-PM-1-21.pdf)
**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](https://www.edgepedia.org/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](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/sigma_algebras.html) There are also subsets of \( (0,1] \) that are not Borel sets, though they cannot be described explicitly.[3](https://personal.math.vt.edu/daymv/class_homepages/6105Notes.pdf)
## 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](https://personal.math.vt.edu/daymv/class_homepages/6105Notes.pdf) • [7](https://www.formalcalculus.com/topics/sigma-algebras) 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](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/measures.html)
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](https://www.edgepedia.org/lebesgue-measure) does support such a measure.[6](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html) 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](https://www.edgepedia.org/banach-tarski-paradox).[2](https://www.maths.dur.ac.uk/users/clare.wallace/Prob1/01-axioms.html) Restricting to a σ-algebra is therefore the device that keeps probability and measure well defined.[9](https://philsci-archive.pitt.edu/16121/1/Interpretive%20Implications%20of%20the%20Sample%20Space%20-%20d4.pdf)
## 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](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/measures.html) • [4](https://www.dpmms.cam.ac.uk/study/II/Probability%2BMeasure/2019-2020/lecture-notes-PM-1-21.pdf) 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](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/measures.html)
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](https://galton.uchicago.edu/~lalley/Courses/381/measure.pdf) 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](https://www.math.toronto.edu/almut/MAT1000/LL-1.pdf) • [10](https://galton.uchicago.edu/~lalley/Courses/381/measure.pdf)
## 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](https://www.mat.univie.ac.at/~mathias/Advanced_Probability.pdf) 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](https://maths.dur.ac.uk/users/clare.wallace/MF/Chapter4.html)
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](https://maths.dur.ac.uk/users/clare.wallace/MF/Chapter4.html)
## Historical origin
The σ-algebra concept emerged from the measure theory built around 1900. In 1898 [Émile Borel](https://www.edgepedia.org/emile-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](https://www.treccani.it/enciclopedia/la-seconda-rivoluzione-scientifica-matematica-e-logica-la-teoria-della-misura_(Storia-della-Scienza)/) 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](https://plato.stanford.edu/entries/settheory-early/)
## References
1. [σ-algebras, Donald Robertson, University of Manchester lecture notes](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/sigma_algebras.html)
2. [1 Axioms of probability, Probability 1 lecture notes, Clare Wallace, Durham University](https://www.maths.dur.ac.uk/users/clare.wallace/Prob1/01-axioms.html)
3. [Unit I: Measures and Sigma-Fields, Virginia Tech 6105 notes](https://personal.math.vt.edu/daymv/class_homepages/6105Notes.pdf)
4. [Probability and Measure lecture notes, DPMMS, University of Cambridge](https://www.dpmms.cam.ac.uk/study/II/Probability%2BMeasure/2019-2020/lecture-notes-PM-1-21.pdf)
5. [Measure and probability: Part I, N. Sanyal, University of Texas at El Paso](https://www.math.utep.edu/faculty/nsanyal/blog/measure_and_prob_1.html)
6. [Probability spaces and Random Variables, Gautam Iyer, Carnegie Mellon University](https://www.math.cmu.edu/~gautam/c/2026-425/notes/probability-spaces.html)
7. [Sigma-Algebras & Measures, formalCalculus](https://www.formalcalculus.com/topics/sigma-algebras)
8. [Measures, Donald Robertson, University of Manchester lecture notes](https://personalpages.manchester.ac.uk/staff/donald.robertson/teaching/23-24/41021/notes/measures.html)
9. [Interpretive Implications of the Sample Space, PhilSci Archive](https://philsci-archive.pitt.edu/16121/1/Interpretive%20Implications%20of%20the%20Sample%20Space%20-%20d4.pdf)
10. [Measure Theory notes, Steve Lalley, University of Chicago](https://galton.uchicago.edu/~lalley/Courses/381/measure.pdf)
11. [Measure and Integration lecture notes, University of Toronto (MAT1000 notes by A. Burchard)](https://www.math.toronto.edu/almut/MAT1000/LL-1.pdf)
12. [Advanced Probability lecture notes, Mathias Beiglböck, University of Vienna](https://www.mat.univie.ac.at/~mathias/Advanced_Probability.pdf)
13. [Chapter 4: Filtrations, Conditional Expectation, and Martingales, Mathematical Finance notes, Clare Wallace, Durham University](https://maths.dur.ac.uk/users/clare.wallace/MF/Chapter4.html)
14. [La seconda rivoluzione scientifica: matematica e logica. La teoria della misura, Enciclopedia Italiana Treccani](https://www.treccani.it/enciclopedia/la-seconda-rivoluzione-scientifica-matematica-e-logica-la-teoria-della-misura_(Storia-della-Scienza)/)
15. [The Early Development of Set Theory, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/settheory-early/)
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