A ring of sets, also known as a ring of subsets, is a fundamental structure in measure theory consisting of a non-empty collection of subsets of a given set X that is closed under finite unions and set differences, and thus necessarily includes the empty set \emptyset.[1] Equivalently, it can be viewed algebraically as a subring of the power set \mathcal{P}(X) under the operations of symmetric difference A \Delta B = (A \setminus B) \cup (B \setminus A) as addition and intersection A \cap B as multiplication, forming a commutative ring with identity \emptyset for addition.[2] This structure satisfies properties such as closure under finite intersections (derived from differences) and symmetric differences, but does not require the inclusion of X itself.[1] If the ring contains X, it is called an algebra of sets (or field of sets), which is then closed under complements and thus under finite unions and intersections of its members.[2]Rings of sets play a central role in measure theory by providing a domain for defining pre-measures that can be extended to sigma-algebras via Carathéodory's extension theorem, enabling the construction of measures like the Lebesgue measure on \mathbb{R}.[1] A sigma-ring extends this by requiring closure under countable unions, forming the basis for sigma-algebras when X is included, which are essential for countable additivity in integrationtheory.[2] Notable examples include the ring generated by finite unions of intervals on the real line or the collection of all finite subsets of an infinite set, highlighting their utility in approximating measurable sets and functions.[1]
Definition
Measure-theoretic ring
In measure theory, a ring of sets is defined as a non-empty family \mathcal{R} of subsets of a given set X that is closed under finite unions and relative complements (set differences). Specifically, for all A, B \in \mathcal{R}, both A \cup B \in \mathcal{R} and A \setminus B \in \mathcal{R}.[3][4]The inclusion of the empty set \emptyset in \mathcal{R} follows directly from the axioms: selecting any A \in \mathcal{R} and setting B = A yields A \setminus A = \emptyset \in \mathcal{R}.[3][4]Closure under finite unions of any finite number of sets in \mathcal{R} can be established by mathematical induction on the number of sets, using the binary union axiom as the base case and associativity for the inductive step.[3][4]Unlike a \sigma-ring, which extends this structure to closure under countable unions, a ring of sets requires only finite operations, making it suitable for foundational constructions in measure theory before extending to more general measurable structures.[3][4]The relative complement operation, or set difference A \setminus B = \{x \in A \mid x \notin B\}, exemplifies how rings facilitate the subtraction of subsets while preserving membership in the family.[3][4]This measure-theoretic notion aligns with abstract algebra by interpreting symmetric difference as addition and intersection as multiplication, forming a Boolean ring.[3]
Order-theoretic ring
In order theory, a ring of sets is defined as a family \mathcal{R} of subsets of a given set X that is closed under finite unions and finite intersections.[5] Under the partial order of set inclusion, \mathcal{R} forms a distributive lattice, where the join operation corresponds to union and the meet operation to intersection.[5] This structure satisfies the distributive laws, such as A \cap (B \cup C) = (A \cap B) \cup (A \cap C) for all A, B, C \in \mathcal{R}.[6] The inclusion of the empty set \emptyset or the full set X is not required in this definition, allowing for proper sublattices of the power set of X.[5]Every measure-theoretic ring of sets—which is closed under finite unions and set differences—is necessarily an order-theoretic ring, as closure under differences implies closure under intersections via the identity A \cap B = A \setminus (A \setminus B).[7] However, the converse does not hold; there exist order-theoretic rings that are not closed under set differences, such as the lattice of initial segments of a finite totally ordered set.[5]If an order-theoretic ring of sets is additionally closed under complements relative to X (assuming X \in \mathcal{R}), it forms a Boolean algebra, a complemented distributive lattice where every element has a unique complement satisfying A \cup A^c = X and A \cap A^c = \emptyset.[5]The notion of an order-theoretic ring of sets emerged within the development of lattice theory in the early 20th century, building on foundational work in Boolean algebra and abstract structures.[5] Key advancements include Garrett Birkhoff's 1937 representation theorem, which establishes that every distributive lattice is isomorphic to a ring of sets, providing a set-theoretic realization of these abstract orders.[5]
Properties
Basic closure properties
A ring of sets, whether in the measure-theoretic or order-theoretic sense, exhibits several fundamental closure properties that follow directly from its defining axioms. In the measure-theoretic definition, a ring \mathcal{R} is a non-empty collection of subsets of a set X that contains the empty set \emptyset and is closed under finite unions and set differences.[8] These axioms imply closure under finite intersections, as for any A, B \in \mathcal{R}, the intersection A \cap B can be expressed as A \setminus (A \setminus B), where both A \setminus B \in \mathcal{R} and the subsequent difference belong to \mathcal{R}.[3] Similarly, closure under symmetric difference holds, since A \Delta B = (A \setminus B) \cup (B \setminus A), combining two differences via union, both operations preserving membership in \mathcal{R}.[8]Finite unions follow immediately from the axioms by induction: the binary case is given, and for n > 2 sets A_1, \dots, A_n \in \mathcal{R}, the union \bigcup_{i=1}^n A_i is obtained iteratively.[3] In the order-theoretic sense, a ring is defined as closed under finite unions and finite intersections, so these properties hold by definition, and contains the empty set (explicitly required in standard definitions).[8] For both definitions, the collection is closed under finite Boolean combinations, meaning any expression formed by finitely many unions, intersections, and differences (or symmetric differences) of sets in the ring remains in the ring.[3]Every ring of sets contains \emptyset, as it arises as A \setminus A in the measure-theoretic case or as the empty union in the order-theoretic case.[8] However, rings do not necessarily contain the whole space X; for instance, the collection of finite subsets of an infinite X forms a ring under either definition but excludes X itself.[3]
Algebraic interpretations
Rings of sets admit a natural algebraic structure as Boolean rings when equipped with symmetric difference as the addition operation and intersection as the multiplication operation. Specifically, for sets A and B in the ring, define addition by A + B = A \Delta B = (A \setminus B) \cup (B \setminus A), which is the symmetric difference, and multiplication by A \cdot B = A \cap B.[9][10]Under these operations, the collection forms an abelian group with respect to addition, where the zero element is the empty set \emptyset, since A + \emptyset = A for any A, and every element is its own inverse due to the idempotence property A + A = \emptyset.[9] The multiplication is associative and commutative, and it distributes over addition because intersection distributes over symmetric difference: A \cap (B \Delta C) = (A \cap B) \Delta (A \cap C), with commutativity ensuring the reverse.[3] Thus, every ring of sets is a commutative ring with this structure.[3]If the ring includes the universal set X (the set containing all elements under consideration), then X serves as the multiplicative identity, since A \cap X = A for any A.[9] Moreover, the multiplication satisfies idempotence: A \cdot A = A \cap A = A for every A, which is a defining characteristic of Boolean rings.[10] This correspondence justifies the use of "ring" in the measure-theoretic sense, as the set operations mirror the axioms of abstract Boolean rings.[3]In this framework, subrings of a ring of sets, such as the power set of a finite set X, correspond to partitions of X, where each subring consists of the unions of blocks from the partition, forming a smaller Boolean ring closed under the operations.[11] For instance, the full power set of X itself is the complete Boolean ring under these operations.[9]
Examples
Finite and power set examples
The trivial ring consisting solely of the empty set, \{\emptyset\}, forms a ring of sets because the union of the empty set with itself is the empty set, and the set difference of the empty set from itself is also the empty set.[12] Another basic example is the collection \{\emptyset, X\} for any universe X, which is closed under finite unions—since \emptyset \cup X = X and X \cup X = X—and set differences—such as X \setminus \emptyset = X, X \setminus X = \emptyset, and \emptyset \setminus X = \emptyset.[1]The power set \mathcal{P}(X) of any set X, comprising all subsets of X, is a ring of sets as it is closed under finite unions, set differences, and hence intersections.[13] This structure includes both the empty set and X itself, making \mathcal{P}(X) not only a ring but also an algebra of sets.[1] Moreover, \mathcal{P}(X) represents the largest possible ring of sets over the universe X, since any ring of subsets must be contained within \mathcal{P}(X) by definition and the closurepropertiesensure \mathcal{P}(X) satisfies the ring axioms.[1]When X is finite, \mathcal{P}(X) provides a canonical finite example of a ring of sets, containing exactly $2^{|X|} elements and closed under the required operations.[13] More generally, any finite collection of subsets of a finite X that is closed under finite unions and set differences qualifies as a finite ring of sets; for instance, on X = \{1, 2\}, the subcollection \{\emptyset, \{1, 2\}\} is a finite ring distinct from the full power set.[1] Such finite rings illustrate the foundational closure properties without the complexity of infinite structures.For an infinite set X, the collection of all finite subsets of X forms a ring of sets. It is closed under finite unions, as the union of finitely many finite sets is finite, and under set differences, as the difference of finite sets is finite. This ring does not include X itself, distinguishing it from the power set, and highlights how rings can approximate measurable sets in infinite spaces.[1]
Topological and interval examples
In a topological space, the collection of all open sets is closed under arbitrary unions (hence finite unions) and finite intersections. This structure arises naturally from the axioms of topology, where the union of any collection of open sets is open, and the intersection of finitely many open sets is open.[14] However, this family is generally not closed under complements or set differences, unless the topology is discrete.[15]A prominent measure-theoretic example is the family of all finite unions of half-open intervals of the form (a, b] with a, b \in \mathbb{R}. This collection is closed under finite unions and set differences, forming a ring of sets, and serves as a foundational structure in real analysis.[16] For instance, the difference of two such finite unions can be expressed as a finite union of half-open intervals by adjusting endpoints appropriately. The ring generated by open intervals coincides with this family, providing a concrete infinite example distinct from power sets.[17]The Borel \sigma-algebra on \mathbb{R} is generated by iteratively applying countable unions and complements to open intervals, but restricting to finite operations yields a ring consisting of finite unions of half-open intervals.[16] This finite-union subcollection forms a proper ring, as it lacks closure under countable unions, in contrast to the full Borel \sigma-ring. Such rings highlight the transition from finite to countable additivity in measure theory.Under group actions, the family of invariant subsets provides another example; specifically, the subsets of \mathbb{R} invariant under rational translations (sets A such that A + q = A for all q \in \mathbb{Q}) form a measure-theoretic ring, closed under finite unions and differences, since these operations preserve invariance.[18] This ring includes sets like the rationals \mathbb{Q} and the irrationals \mathbb{R} \setminus \mathbb{Q}.In order theory, the upper sets (or upsets) of a partially ordered set (X, \leq) form a collection closed under arbitrary unions and intersections.[19] An upset U \subseteq X satisfies: if x \in U and x \leq y, then y \in U; the union and intersection of upsets are upsets, ensuring the structure. This example connects set collections to lattice theory, where upsets correspond to order ideals in the dual poset.
Constructions
Generating rings from families
The ring generated by a family \mathcal{C} \subseteq \mathcal{P}(X) is defined as the intersection of all rings of subsets of X that contain \mathcal{C}; this ensures it is the smallest such ring.[20]To construct the generated ring explicitly, begin with the collection \mathcal{C}_0 = \mathcal{C} \cup \{\emptyset\} (including the empty set for closure). Iteratively form \mathcal{C}_{n+1} by adjoining to \mathcal{C}_n all finite unions and all set differences A \setminus B where A, B \in \mathcal{C}_n. The generated ring is then \mathcal{R}(\mathcal{C}) = \bigcup_{n=0}^\infty \mathcal{C}_n, which achieves closure under these operations after finitely many steps in practice.[20]If the initial family \mathcal{C} is a semiring (closed under finite intersections and such that the complement of any set in \mathcal{C} relative to another set in \mathcal{C} can be expressed as a finite disjoint union of sets in \mathcal{C}), the generated ring admits a more direct description: it consists precisely of all sets that can be written as finite disjoint unions of elements from \mathcal{C}. To obtain this, start with \mathcal{C} and adjoin all such disjoint unions, which automatically ensures closure under differences via the semiring property (expressing A \setminus B as a disjoint union when B \subseteq A) and under unions by disjointification.[21][22]A concrete example arises on the real line \mathbb{R}, where the family \mathcal{C} of all bounded open intervals (a, b) with a < b generates a ring consisting of all finite disjoint unions of such intervals; for instance, (0,1) \cup (2,3) belongs to the ring, and differences like (0,3) \setminus (1,2) = (0,1) \cup (2,3) preserve this form.[21]When \mathcal{C} is finite, the iterative construction yields a finite ring, as the Boolean combinations (unions and differences) of finitely many generating sets produce only finitely many distinct subsets of X.[20]
Operations on collections of rings
The intersection of an arbitrary family of rings of sets \{\mathcal{R}_i\}_{i \in I} on a common universe X is itself a ring on X. This holds because the intersection contains the empty set (present in each \mathcal{R}_i) and is closed under finite unions and set differences: for any A, B \in \bigcap_i \mathcal{R}_i, both A and B belong to every \mathcal{R}_i, so A \cup B \in \mathcal{R}_i and A \setminus B \in \mathcal{R}_i for all i, hence these operations lie in the intersection.[21][12]A subring of a ring \mathcal{R} on X is any subcollection \mathcal{S} \subseteq \mathcal{R} that forms a ring under the same operations of union and difference. Such subrings inherit closure properties directly from \mathcal{R}, and examples include traces of \mathcal{R} onto subsets of X.[12]The union \bigcup_i \mathcal{R}_i of a family of rings on X is not necessarily a ring, as it may fail to be closed under unions or differences. However, there exists a unique smallest ring containing this union, called the ring generated by the union.[21] If the family \{\mathcal{R}_i\} forms a chain under inclusion (i.e., for any i, j, either \mathcal{R}_i \subseteq \mathcal{R}_j or \mathcal{R}_j \subseteq \mathcal{R}_i), then the union is itself a ring, preserving closure under the operations since any two sets in the union belong to some common \mathcal{R}_k in the chain.[21]For spaces Y and Z with rings \mathcal{R}_Y on Y and \mathcal{R}_Z on Z, the product space X = Y \times Z admits a product ring, which is the ring generated by the rectangles A \times B where A \in \mathcal{R}_Y and B \in \mathcal{R}_Z; this structure is closed under finite unions and differences, facilitating constructions like product measures.[23]
Related structures
Semirings and set algebras
A semiring of sets on a ground set X is a nonempty collection \mathcal{S} of subsets of X that contains the empty set \emptyset and is closed under finite intersections, such that for any A, B \in \mathcal{S}, the set difference A \setminus B can be expressed as a finite union of pairwise disjoint sets from \mathcal{S}.[24] This structure serves as a foundational precursor to rings of sets, providing a partial closure under set operations that facilitates extension to full rings without requiring additive inverses for every element.[25]Any semiring \mathcal{S} generates a ring of sets by taking the smallest ring containing \mathcal{S}, which consists precisely of all finite unions of pairwise disjoint members of \mathcal{S}.[24] This construction ensures closure under finite unions and differences, as intersections of such unions remain expressible in the same form, and the resulting collection forms a ring.[25]In contrast, an algebra of sets on X is a ring of sets that is additionally closed under complements relative to X, meaning that if A \in \mathcal{A}, then X \setminus A \in \mathcal{A}.[26] Such a structure automatically contains X (as the complement of \emptyset) and is thus equivalent to a field of sets, providing a Boolean algebra framework for the power set operations.[26] Unlike rings, which may not include complements and thus allow for "partial" families without reference to the full universe X, algebras enforce completeness under complementation, enabling symmetric difference and full Boolean closure.[26]For instance, on the real line \mathbb{R}, the collection of all half-open intervals of the form (a, b] with a < b forms a semiring, as their intersections are again half-open intervals or empty, and differences decompose into finitely many disjoint such intervals.[25] The ring generated by this semiring comprises all finite unions of these half-open intervals, which is closed under the ring operations but not under complements relative to \mathbb{R}.[27]
Sigma-rings and fields of sets
A sigma-ring of sets is a ring of sets that is additionally closed under countable unions. That is, if \{A_n\}_{n=1}^\infty \subset \mathcal{R}, then \bigcup_{n=1}^\infty A_n \in \mathcal{R}.[8] Since a ring is already closed under finite unions and set differences, closure under countable unions implies closure under countable intersections as well, via the relation \bigcap_{n=1}^\infty A_n = A_1 \setminus \bigcup_{n=2}^\infty (A_1 \setminus A_n) for nested sequences or more generally using relative complements.[8]Unlike sigma-algebras, sigma-rings are not required to contain the underlying universal set X or to be closed under absolute complements X \setminus A. This distinction allows sigma-rings to model structures where the total space is "infinite" or not included, such as in non-sigma-finite measures. Rings of sets, including sigma-rings, form Boolean rings under the operations of symmetric difference (as addition) and intersection (as multiplication), where every element is idempotent.[7]A field of sets, also known as an algebra of sets, is a ring of sets that contains the universal set X and is closed under complements relative to X, meaning if A \in \mathcal{F}, then X \setminus A \in \mathcal{F}. This closure ensures that fields are Boolean rings with unity, where the unit element is X. Consequently, fields are closed under finite unions, intersections, and differences, forming a complete Boolean algebra structure on the power set restricted to \mathcal{F}.[28][7]A sigma-field of sets extends this to countable operations: it is a field closed under countable unions (and thus countable intersections and complements). This is precisely a sigma-algebra, the standard structure for measurable sets in measure theory. Sigma-fields always contain X and the empty set, distinguishing them from general sigma-rings.[8][7]For an example contrasting these structures, consider the real line \mathbb{R}. The Borel sigma-algebra \mathcal{B}(\mathbb{R}), generated by the open sets, is a sigma-field containing \mathbb{R} and closed under countable operations, enabling the definition of Lebesgue measure on all Borel sets. In contrast, the collection of all countable subsets of \mathbb{R} forms a sigma-ring: it includes the empty set, is closed under countable unions (yielding countable sets) and relative differences (also countable), but excludes \mathbb{R} itself since \mathbb{R} is uncountable, so it is not a sigma-field.[29]
Applications
Role in measure construction
Rings of sets play a fundamental role in the construction of measures in measure theory, serving as the domain for pre-measures that can be extended to sigma-algebras. A pre-measure on a ring \mathcal{R} of subsets of a set X is a countably additive set function \mu: \mathcal{R} \to [0,\infty] satisfying \mu(\emptyset) = 0. This countably additivity ensures that for any countable disjoint union of sets in \mathcal{R}, the measure of the union equals the sum of the measures, providing a foundation for extending the function while preserving additivity properties.From such a pre-measure, an outer measure \mu^* is defined on the power set of X by\mu^*(E) = \inf \left\{ \sum \mu(A_i) : E \subset \bigcup A_i, \, A_i \in \mathcal{R} \right\},where the infimum is taken over all countable covers of E by sets from the ring. This outer measure extends the pre-measure in a minimal way, approximating the "size" of arbitrary sets E using elements from \mathcal{R}.Carathéodory's extension theorem then identifies the measurable sets as those E satisfying the Carathéodory condition: for any set T \subseteq X, \mu^*(T) = \mu^*(T \cap E) + \mu^*(T \setminus E).[30] The collection of such measurable sets forms a sigma-algebra containing \mathcal{R}, and the restriction of \mu^* to this sigma-algebra yields a measure that agrees with \mu on \mathcal{R}. For sigma-finite pre-measures, this extension is unique.This framework, introduced by Constantin Carathéodory in 1914, provided a rigorous method to construct Lebesgue measure by starting from the ring of finite unions of intervals on the real line.[30]
Uses in probability and analysis
In probability theory, rings of sets provide a natural domain for defining finitely additive probability measures, which satisfy additivity only for finite disjoint unions rather than countable ones. A finitely additive measure \mu on a ring \mathcal{R} of subsets of a space \Omega is a function \mu: \mathcal{R} \to [0,1] with \mu(\emptyset) = 0 and \mu(A \cup B) = \mu(A) + \mu(B) whenever A, B \in \mathcal{R} are disjoint, often normalized so that \mu(\Omega) = 1 if \mathcal{R} is an algebra containing \Omega. These measures arise in foundational work on subjective probability and are essential for modeling scenarios where countable additivity leads to paradoxes or incompleteness, such as in infinite state spaces without additional structure.[31]A prominent application occurs in the construction of stochastic processes via the Kolmogorov extension theorem. Here, the ring of cylinder sets in an infinite product space, such as \mathbb{R}^{\mathbb{N}}, consists of sets defined by constraints on finitely many coordinates, forming a ring closed under finite unions and differences. Finite-dimensional distributions specify consistent finitely additive measures on this ring, which the theorem extends to a countably additive probability measure on the generated \sigma-algebra, enabling the rigorous definition of processes like Brownian motion. This framework highlights rings' role in bridging finite approximations to infinite-dimensional probability spaces.[32][33]In analysis, rings of sets underpin the Daniell integral, a method for defining integrals through positive linear functionals on lattices of functions without presupposing a measure. Starting from a functional I satisfying linearity, monotonicity, and positivity on a space of bounded functions, the integrable functions form a lattice, and the induced Daniell-measurable sets constitute a \sigma-ring on which I defines a measure \mu(E) = I(\chi_E) for characteristic functions \chi_E. This yields the Daniell-Stone theorem, equating Daniell integrals to Lebesgue integrals with respect to \mu, and proves useful in abstract settings like potential theory or when extending integrals to non-\sigma-finite domains. Rings thus facilitate measure construction in cases where traditional \sigma-algebras prove restrictive, such as non-\sigma-finite measures like the counting measure on uncountable sets, where finite subsets have finite measure but the whole space does not, allowing controlled extensions via pre-measures on the ring.[34]In harmonic analysis, rings comprising sets invariant under group actions support the definition of convolution measures that commute with the group operation. For a locally compact group G acting on a space, the ring of G-invariant subsets admits measures \mu such that \mu(gA) = \mu(A) for g \in G and A in the ring, enabling convolutions \mu * \nu to preserve invariance and facilitating Fourier transforms on non-abelian groups. This structure is crucial for decomposing representations and studying invariant subspaces in L^1(G), with applications to ergodic theory and spectral analysis.[35]