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Borel set

In mathematics, particularly in measure theory and topology, a Borel set is an element of the Borel σ-algebra on a topological space, defined as the smallest σ-algebra that contains all open sets of the space.^[1] This σ-algebra is generated by iteratively applying countable unions, countable intersections, and complements starting from the open sets, resulting in a hierarchy of sets including open sets (denoted \mathbf{\Sigma}_1^0), closed sets (\mathbf{\Pi}_1^0), F_\sigma sets (countable unions of closed sets), G_\delta sets (countable intersections of open sets), and more complex levels up to the full Borel hierarchy.^[2] Named after the French mathematician Émile Borel, who introduced the concept in his 1898 book Leçons sur la théorie des fonctions, these sets form a foundational structure for assigning measures to subsets of spaces like the real line \mathbb{R}^n.^[3] Borel sets play a central role in real analysis and probability theory, as they provide a rich yet manageable collection of "measurable" sets that includes all open and closed sets while avoiding the paradoxes associated with measuring arbitrary subsets of the continuum.^[4] The Lebesgue measure, a cornerstone of modern integration theory, is defined precisely on the Borel σ-algebra of \mathbb{R}^n, extending naturally to the completion that includes all Lebesgue measurable sets, though not all Lebesgue measurable sets are Borel (there are $2^{2^{\aleph_0}} Lebesgue measurable sets but only continuum many Borel sets).^[2] In probability, Borel sets serve as the standard σ-algebra for sample spaces in continuous models, enabling the definition of random variables and distributions on spaces like \mathbb{R}.^[4] Their construction ensures closure under the operations needed for limits and convergence theorems, such as those involving pointwise limits of measurable functions, where continuity points form G_\delta sets.^[2]

Generating the Borel σ-algebra

Definition and construction

In a topological space (X, \tau), where \tau denotes the collection of open sets, a \sigma-algebra on X is a family of subsets of X that contains the empty set and X itself, and is closed under complementation relative to X and countable unions (hence also countable intersections). The Borel \sigma-algebra \mathcal{B}(X) (or B(X)) is defined as the smallest \sigma-algebra on X containing all open sets in \tau, or equivalently, the \sigma-algebra generated by \tau.^[5] Since the complement of an open set is closed, \mathcal{B}(X) is also generated by the closed sets of X.^[5] The Borel sets, which form \mathcal{B}(X), consist of all subsets of X that can be obtained from the open sets through countable applications of unions, intersections, and complements. This construction proceeds via transfinite iteration over the ordinals up to the first uncountable ordinal \omega_1. The process stratifies the Borel sets into a hierarchy of classes \mathbf{\Sigma}^0_\alpha and \mathbf{\Pi}^0_\alpha for each countable ordinal \alpha < \omega_1.^[5] The hierarchy begins with the base level: \mathbf{\Sigma}^0_1(X) comprises the open sets of X, while \mathbf{\Pi}^0_1(X) consists of the closed sets (complements of open sets). For a successor ordinal \alpha + 1 < \omega_1, \mathbf{\Sigma}^0_{\alpha+1}(X) is the collection of all countable unions of sets from \bigcup_{\beta < \alpha+1} \mathbf{\Pi}^0_\beta(X), and \mathbf{\Pi}^0_{\alpha+1}(X) is the collection of all countable intersections of sets from \bigcup_{\beta < \alpha+1} \mathbf{\Sigma}^0_\beta(X). At a limit ordinal \lambda < \omega_1, the classes are defined as \mathbf{\Sigma}^0_\lambda(X) = \bigcup_{\beta < \lambda} \mathbf{\Sigma}^0_\beta(X) and \mathbf{\Pi}^0_\lambda(X) = \bigcup_{\beta < \lambda} \mathbf{\Pi}^0_\beta(X). The full Borel \sigma-algebra is then the union \mathcal{B}(X) = \bigcup_{\alpha < \omega_1} \mathbf{\Sigma}^0_\alpha(X) = \bigcup_{\alpha < \omega_1} \mathbf{\Pi}^0_\alpha(X).^[5]

Example in Euclidean space

In Euclidean space \mathbb{R}, the standard topology is generated by the collection of all open intervals (a, b) where a < b, and any open set is a countable union of such disjoint open intervals.^[6] The Borel \sigma-algebra \mathcal{B}(\mathbb{R}) is then the smallest \sigma-algebra containing these open sets, so all open sets belong to \mathcal{B}(\mathbb{R}) by definition.^[6] Closed sets in \mathbb{R} are complements of open sets and thus also Borel, forming the class \Pi_1^0 in the Borel hierarchy. For instance, singletons \{x\} for x \in \mathbb{R} are closed (as complements of the open set \mathbb{R} \setminus \{x\}) and hence Borel.^[6] The middle-thirds Cantor set C \subset [0,1], constructed by iteratively removing middle open intervals from [0,1], is a closed set (as the intersection of a decreasing sequence of closed sets) and therefore Borel with hierarchy level \Pi_1^0.^[7] Countable unions of closed sets, known as F_\sigma sets or \Sigma_2^0 sets, are also Borel. The set of rational numbers \mathbb{Q} exemplifies this: enumerate \mathbb{Q} = \{q_n : n \in \mathbb{N}\}, so \mathbb{Q} = \bigcup_{n=1}^\infty \{q_n\}, a countable union of closed singletons, making \mathbb{Q} Borel at level \Sigma_2^0.^[2] Countable intersections of open sets, or G_\delta sets at level \Pi_2^0, belong to the Borel \sigma-algebra as well. The irrationals \mathbb{R} \setminus \mathbb{Q} form such a set: for an enumeration \mathbb{Q} = \{q_n : n \in \mathbb{N}\}, \mathbb{R} \setminus \mathbb{Q} = \bigcap_{n=1}^\infty (\mathbb{R} \setminus \{q_n\}), a countable intersection of open sets, confirming its Borel status.^[8] In \mathbb{R}^n equipped with the product topology, open sets are countable unions of products of open intervals, such as \prod_{i=1}^n (a_i, b_i), which generate the Borel \sigma-algebra \mathcal{B}(\mathbb{R}^n).^[9] This Borel \sigma-algebra serves as the foundation for the Lebesgue \sigma-algebra, obtained by completing \mathcal{B}(\mathbb{R}^n) with respect to Lebesgue measure, where every Lebesgue measurable set differs from a Borel set by a set of measure zero.^[9]

Borel Hierarchy and Properties

The Borel hierarchy

The Borel hierarchy provides a stratified classification of the Borel sets in a topological space, organizing them according to their descriptive complexity using transfinite iterations of countable unions and complements, indexed by countable ordinals up to the first uncountable ordinal \omega_1.^[10]^[11] This hierarchy is fundamental in descriptive set theory, distinguishing sets based on the minimal ordinal level required to generate them from open sets.^[10] The hierarchy is denoted using boldface letters: for each countable ordinal \alpha < \omega_1, the classes \boldsymbol{\Sigma}^0_\alpha, \boldsymbol{\Pi}^0_\alpha, and \boldsymbol{\Delta}^0_\alpha are defined recursively. The base levels are \boldsymbol{\Sigma}^0_1, the class of open sets, and \boldsymbol{\Pi}^0_1, the class of closed sets, with \boldsymbol{\Delta}^0_1 = \boldsymbol{\Sigma}^0_1 \cap \boldsymbol{\Pi}^0_1 consisting of clopen sets.^[11] For successor ordinals \alpha = \beta + 1, \boldsymbol{\Sigma}^0_\alpha consists of all countable unions of sets from \boldsymbol{\Pi}^0_\beta, while \boldsymbol{\Pi}^0_\alpha consists of all countable intersections of sets from \boldsymbol{\Sigma}^0_\beta (equivalently, complements of sets in \boldsymbol{\Sigma}^0_\alpha); the diagonal class is \boldsymbol{\Delta}^0_\alpha = \boldsymbol{\Sigma}^0_\alpha \cap \boldsymbol{\Pi}^0_\alpha.^[10]^[11] At limit ordinals \lambda, the classes are defined as \boldsymbol{\Sigma}^0_\lambda = \bigcup_{\alpha < \lambda} \boldsymbol{\Sigma}^0_\alpha and \boldsymbol{\Pi}^0_\lambda = \bigcup_{\alpha < \lambda} \boldsymbol{\Pi}^0_\alpha.^[11] Early levels include \boldsymbol{\Sigma}^0_2, the F_\sigma sets (countable unions of closed sets), and \boldsymbol{\Pi}^0_2, the G_\delta sets (countable intersections of open sets).^[10] The hierarchy is complete in the sense that the Borel σ-algebra is precisely the union \bigcup_{\alpha < \omega_1} \boldsymbol{\Sigma}^0_\alpha = \bigcup_{\alpha < \omega_1} \boldsymbol{\Pi}^0_\alpha = \bigcup_{\alpha < \omega_1} \boldsymbol{\Delta}^0_\alpha.^[11] In uncountable Polish spaces, the hierarchy is strict: for each \alpha < \omega_1, \boldsymbol{\Sigma}^0_\alpha \subsetneq \boldsymbol{\Delta}^0_{\alpha+1} \subsetneq \boldsymbol{\Sigma}^0_{\alpha+1} and similarly for the \boldsymbol{\Pi} classes, with no collapsing of levels.^[10] A key property of Borel sets within this hierarchy is captured by the perfect set theorem: every uncountable Borel set in a Polish space contains a perfect subset, which is closed, has no isolated points, and is homeomorphic to the Cantor set.^[11] This theorem implies that Borel sets are either countable or have cardinality of the continuum, highlighting their regularity compared to more complex descriptive classes.^[10]

Key properties of Borel sets

Borel sets form a σ-algebra, and thus are closed under complements and countable unions; consequently, they are also closed under countable intersections and finite unions. This closure under countable operations distinguishes the Borel σ-algebra as the smallest collection containing all open sets and stable under these set-theoretic constructions.^[12] Borel sets constitute a proper subclass of the analytic sets, which are continuous images of Borel sets and include all Borel sets as a special case.^[12] A fundamental universal property of the Borel σ-algebra on a topological space is that it is the smallest σ-algebra rendering all continuous real-valued functions measurable; any coarser σ-algebra would fail to make some continuous function measurable, as preimages of open intervals under continuous maps are open sets.^[12] On the real line \mathbb{R}, the Borel σ-algebra \mathcal{B}(\mathbb{R}) has cardinality |\mathcal{B}(\mathbb{R})| = \mathfrak{c}, the cardinality of the continuum, even though the power set of \mathbb{R} has cardinality $2^{\mathfrak{c}}; this follows from the Borel hierarchy, where open sets number \mathfrak{c} many, and transfinite iterations of countable unions, intersections, and complements up to length \omega_1 yield at most \mathfrak{c}^{\aleph_1} = \mathfrak{c} sets overall.^[12] In metric spaces such as \mathbb{R}^n equipped with Lebesgue measure, every Borel set is Lebesgue measurable, as the Borel sets are generated from open sets via countable operations that preserve measurability under the standard extension of Lebesgue measure. Furthermore, every Borel set in a Polish space possesses the property of Baire, meaning it differs from an open set by a meager set (a countable union of nowhere dense sets).

Standard Borel Spaces

Definition and structure

A standard Borel space is a measurable space (X, \mathcal{B}(X)), where X is a Polish space—that is, a separable completely metrizable topological space—and \mathcal{B}(X) denotes the Borel \sigma-algebra on X generated by its open sets. More generally, any measurable space Borel isomorphic to such a pair (X, \mathcal{B}(X)) is also called a standard Borel space. Standard Borel spaces are characterized by having a countably generated \sigma-algebra that separates points. A Borel isomorphism between measurable spaces (X, \mathcal{B}_X) and (Y, \mathcal{B}_Y) is a bijection f: X \to Y such that f and its inverse f^{-1} are both measurable functions, meaning they map Borel sets in their respective \sigma-algebras to Borel sets. This isomorphism preserves the Borel structure, allowing standard Borel spaces to be characterized up to their cardinality.^[12] Prominent examples of standard Borel spaces include the Euclidean spaces \mathbb{R}^n equipped with the standard topology, the Cantor space $2^\mathbb{N} consisting of infinite binary sequences with the product topology, and the separable Hilbert space \ell^2 of square-summable real sequences with its norm topology. A key structural result is that all uncountable standard Borel spaces are Borel isomorphic to one another, implying they share the same measurable properties despite differing underlying topologies.^[12] Uncountable standard Borel spaces exhibit an atomless structure in their \sigma-algebras, meaning no minimal nonempty Borel sets exist—every nonempty Borel set can be partitioned into two disjoint nonempty Borel subsets. This atomless quality, combined with their uniform isomorphism class, positions standard Borel spaces as the canonical setting for classical descriptive set theory, where Borel sets and their hierarchies are analyzed systematically.^[12]

Kuratowski theorems

Kuratowski's theorem provides a key isomorphism classification of standard Borel spaces. Standard Borel spaces are, up to Borel isomorphism, either finite or countable discrete spaces, or isomorphic to the Baire space (\mathbb{N}^\mathbb{N}, \mathcal{B}), where \mathcal{B} denotes the Borel \sigma-algebra. Equivalently, uncountable standard Borel spaces are all Borel-isomorphic to the real line (\mathbb{R}, \mathcal{B}(\mathbb{R})). This result underscores the uniformity of uncountable standard spaces, implying that their cardinality is the continuum, and any two such spaces share the same structural properties under measurable bijections that are bimeasurable.^[12] A proof of this theorem relies on the Souslin operation, which constructs analytic sets as projections of Borel sets in product spaces. To establish the isomorphism, one shows that any standard Borel space embeds into a Polish space via a measurable map, using analytic sets to approximate subsets and ensure the necessary separability. The Souslin theorem (that analytic sets with the perfect set property are Borel under certain conditions) aids in verifying the isomorphism.^[13] Historically, Stanisław Ulam contributed to the understanding of these characterizations by exploring properties of analytic sets in early developments of descriptive set theory, particularly in showing results like the perfect set theorem for analytic sets, which complements the topological conditions in descriptive set theory.^[13]

Non-Borel Sets

Existence and construction

The existence of non-Borel sets follows from a cardinality argument. The power set \mathcal{P}(\mathbb{R}) of the real numbers has cardinality $2^{\mathfrak{c}}, where \mathfrak{c} = |\mathbb{R}| denotes the cardinality of the continuum, while the Borel \sigma-algebra \mathcal{B}(\mathbb{R}) has cardinality \mathfrak{c}. Consequently, there are $2^{\mathfrak{c}} many subsets of \mathbb{R} that are not Borel sets.^[14] Explicit constructions of non-Borel sets rely on the axiom of choice (AC). One such example is the Vitali set, introduced by Giuseppe Vitali in 1905. Consider the equivalence relation on \mathbb{R} defined by x \sim y if and only if x - y \in \mathbb{Q}. The equivalence classes are the cosets x + \mathbb{Q}. Using AC, select one representative from each coset intersected with the interval [0,1), yielding a set V \subset [0,1). The set V is non-Lebesgue measurable, as its countable rational translates cover $[0,1]without overlap but lead to a contradiction under the additivity of [Lebesgue measure](/page/Lebesgue_measure). Since all Borel sets are Lebesgue measurable,V$ cannot be Borel.^[15] Another construction, due to Felix Bernstein in 1905, yields a Bernstein set B \subset \mathbb{R}, which intersects every uncountable closed subset of \mathbb{R} (every perfect set), as does its complement \mathbb{R} \setminus B. To build B, enumerate all perfect subsets of \mathbb{R} as \{P_\alpha : \alpha < \mathfrak{c}\} via a well-ordering of type \mathfrak{c}. Proceed by transfinite recursion: at stage \alpha, select two distinct points p_\alpha, q_\alpha \in P_\alpha \setminus \bigcup_{\beta < \alpha} \{p_\beta, q_\beta\} using AC, assigning p_\alpha to B and q_\alpha to \mathbb{R} \setminus B. The resulting B lacks the Baire property, meaning it is neither meager nor comeager in any non-empty open set. As all Borel sets possess the Baire property, B is non-Borel.^[15] The Sierpiński set, introduced by Wacław Sierpiński in 1924, provides another example under the continuum hypothesis. It is an uncountable subset S \subset \mathbb{R} such that S \cap N is at most countable for every Lebesgue null set N \subset \mathbb{R}. The existence of such sets is independent of ZFC; the construction employs AC and transfinite recursion by enumerating all null G_\delta sets (of which there are \mathfrak{c} many) and selecting points at each stage from the complement of the union of previously enumerated null G_\delta sets, ensuring the intersection property. Such a set S is non-Lebesgue measurable, hence non-Borel.^[16]

Implications for measure theory

The Lebesgue σ-algebra on \mathbb{R} is defined as the completion of the Borel σ-algebra with respect to Lebesgue measure, incorporating all Borel sets along with all subsets of Borel null sets and their complements.^[17] This structure ensures that every Borel set is Lebesgue measurable, but the Lebesgue σ-algebra is strictly larger, with cardinality exceeding that of the continuum, allowing certain non-Borel sets to be Lebesgue measurable precisely when they differ from Borel sets by null sets.^[17] The axiom of choice enables the construction of non-Borel sets that are Lebesgue measurable, such as certain analytic sets, but it also permits non-measurable pathologies that lie outside this completion.^[18] A prominent implication arises from the Vitali set, a non-Lebesgue measurable set whose construction, as briefly referenced earlier, relies on selecting representatives from rational equivalence classes and obstructs any translation-invariant extension of Lebesgue measure to all subsets of \mathbb{R}.[](https://e.math.cornell.edu/people/belk/measure theory/NonMeasurableSets.pdf) If the Vitali set had positive Lebesgue measure, its countable disjoint rational translates would cover [0,1] with infinite measure, a contradiction; if zero measure, they would cover with zero measure, another contradiction. This non-measurability underscores the limitations of the Borel σ-algebra in measure theory: while Borel sets generate the Lebesgue measure and suffice for regular purposes, the full power set of \mathbb{R} cannot admit a complete translation-invariant measure, necessitating the restricted domain of the Lebesgue σ-algebra for consistency.^[19] In probability theory, Borel sets provide the foundational σ-algebra for defining continuous distributions, as the induced measures from cumulative distribution functions are supported on the Borel σ-algebra and align with Lebesgue measure for absolutely continuous cases.^[20] For standard random variables with continuous densities, Borel measurability ensures well-defined expectations and integrals without invoking the larger Lebesgue σ-algebra, avoiding complications from null sets.^[20] Nonetheless, in advanced stochastic processes—such as those involving irregular paths—non-Borel pathologies demand explicit verification of Borel measurability to maintain theoretical coherence and prevent inconsistencies in sample path properties.^[19] Non-Borel sets create a critical bridge to descriptive set theory, where the analytic sets (continuous images of Borel sets) and coanalytic sets extend beyond the Borel hierarchy, yet Suslin's theorem guarantees that every analytic set is Lebesgue measurable.^[18] This theorem, established in 1917, resolves a key gap by confirming the measurability of these projective sets under Lebesgue measure, thereby extending the applicability of measure theory to broader classes without encountering non-measurable exceptions in this regime.^[18] Consequently, while non-Borel sets challenge the universality of Borel-generated measures, such results affirm the robustness of Lebesgue measurability for descriptively definable sets in probability and analysis.^[18]

Alternative Definitions

Equivalent characterizations

In a topological space X, the Borel \sigma-algebra \mathcal{B}(X) can equivalently be defined as the smallest \sigma-algebra containing all closed subsets of X. This follows from the fact that the closed sets are precisely the complements of the open sets, and any \sigma-algebra closed under complements and containing the opens must contain the closeds, and vice versa.^[21]^[2] Another equivalent characterization is that \mathcal{B}(X) is the smallest \sigma-algebra on X such that every continuous function f: X \to \mathbb{R} is measurable with respect to \mathcal{B}(X) and the Borel \sigma-algebra on \mathbb{R}. Indeed, the preimage under any continuous f of an open set in \mathbb{R} is open in X, so lies in \mathcal{B}(X); conversely, the open sets in X are generated as such preimages from continuous functions to \mathbb{R}.^[22]^[23] In the case of a locally compact Hausdorff space X, \mathcal{B}(X) is also the \sigma-algebra generated by the compact subsets of X. Here, every compact set is closed, and in such spaces, the compact sets form a basis for generating the topology via their complements and unions, yielding the same \sigma-algebra as the opens.^[24]^[25] Finally, in Polish spaces, a set is Borel if and only if it is both analytic and co-analytic, where a set is analytic if it is the projection of a Borel set in the product space X \times \mathbb{N}^\mathbb{N}. This characterization arises from the Suslin theorem, which identifies Borel sets within the projective hierarchy as those analytic sets whose complements are also analytic.^[26]^[27]

Non-equivalent definitions

The Baire σ-algebra on a topological space is defined as the σ-algebra generated by the zero sets of continuous real-valued functions, which are precisely the closed G_δ sets. In general topological spaces, this σ-algebra is contained in the Borel σ-algebra. In second countable spaces, such as the real line, the Baire σ-algebra coincides with the Borel σ-algebra.^[28] The Lebesgue σ-algebra on ℝ is the completion of the Borel σ-algebra with respect to Lebesgue measure, consisting of all sets of the form B ∪ N, where B is Borel and N is a subset of a Borel null set. This σ-algebra properly contains the Borel σ-algebra, as there exist non-Borel Lebesgue measurable sets, many of which are null sets not in the Borel hierarchy. For example, a Vitali set, when modified to have measure zero, exemplifies a non-Borel null set included in the Lebesgue σ-algebra. The completion process ensures that every subset of a null Borel set is measurable with measure zero, expanding beyond the Borel class while preserving measure properties.^[29]^[30] In descriptive set theory, the projective hierarchy begins with the analytic sets, denoted Σ¹₁, defined as the continuous images of Borel sets from Polish spaces such as the Baire space ω^ω. This class properly contains the Borel sets, as analytic sets are closed under continuous images and countable unions but include non-Borel examples, such as the set of all well-founded trees or certain pathological subsets constructed via the axiom of choice. The existence of non-Borel analytic sets was first demonstrated by Sierpiński in 1920, highlighting how projection or continuous preimages can escape the Borel hierarchy while remaining "definable" in a broader sense. Higher levels of the projective hierarchy, like Π¹₁ coanalytic sets, further extend this structure.^[31]^[32] Early 20th-century proposals for classes resembling Borel sets included variants by Felix Hausdorff, who in 1914 introduced a transfinite hierarchy starting from closed sets and using countable unions and intersections, which coincides with the standard Borel class in metric spaces but was motivated by moment functionals in representing measures on intervals. Hausdorff's approach in "Grundzüge der Mengenlehre" emphasized reducible sets like F_σ ∩ G_δ, forming subclasses distinct from full Borel sets in general topologies, and his 1916 work on the cardinality of Borel sets explored how such generations yield sets of cardinality at most the continuum. These historical definitions, while equivalent in separable metric spaces, differ in non-separable or pathological spaces, generating σ-algebras that may exclude certain compact-generated sets.^[33]

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### Summary of Hausdorff’s Historical Variants or Proposals Related to Borel Sets or Sigma-Algebras