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In mathematics, the Borel sets of a topological space are a particular class of "well-behaved" subsets of that space. For example, whereas an arbitrary subset of the real numbers might fail to be Lebesgue measurable, every Borel set of reals is universally measurable. Which sets are Borel can be specified in a number of equivalent ways. Borel sets are named after Émile Borel.

The most usual definition goes through the notion of a σ-algebra, which is a collection of subsets of a topological space that contains both the empty set and the entire set , and is closed under countable union and countable intersection.[a]

Then we can define the Borel σ-algebra over to be the smallest σ-algebra containing all open sets of .[b] A Borel subset of is then simply an element of this σ-algebra.

Borel sets are important in measure theory, since any measure defined on the open sets of a space, or on the closed sets of a space, must also be defined on all Borel sets of that space. Any measure defined on the Borel sets is called a Borel measure. Borel sets and the associated Borel hierarchy also play a fundamental role in descriptive set theory.

In some contexts, Borel sets are defined to be generated by the compact sets of the topological space, rather than the open sets. The two definitions are equivalent for many well-behaved spaces, including all Hausdorff σ-compact spaces, but can be different in more pathological spaces.

Generating the Borel algebra

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In the case that is a metric space, the Borel algebra in the first sense may be described generatively as follows.

For a collection of subsets of (that is, for any subset of the power set of ), let

  • be all countable unions of elements of
  • be all countable intersections of elements of

Now define by transfinite induction a sequence , where is an ordinal number, in the following manner:

  • For the base case of the definition, let be the collection of open subsets of .
  • If is not a limit ordinal, then has an immediately preceding ordinal . Let
  • If is a limit ordinal, set

The claim is that the Borel algebra is , where is the first uncountable ordinal number. That is, the Borel algebra can be generated from the class of open sets by iterating the operation to the first uncountable ordinal.

To prove this claim, any open set in a metric space is the union of an increasing sequence of closed sets. In particular, complementation of sets maps into itself for any limit ordinal ; moreover if is an uncountable limit ordinal, is closed under countable unions.

For each Borel set , there is some countable ordinal such that can be obtained by iterating the operation over . However, as varies over all Borel sets, will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is , the first uncountable ordinal.

The resulting sequence of sets is termed the Borel hierarchy.

Example

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An important example, especially in the theory of probability, is the Borel algebra on the set of real numbers. It is the algebra on which the Borel measure is defined. Given a real random variable defined on a probability space, its probability distribution is by definition also a measure on the Borel algebra.

The Borel algebra on the reals is the smallest σ-algebra on that contains all the intervals.

In the construction by transfinite induction, it can be shown that, in each step, the number of sets is, at most, the cardinality of the continuum. So, the total number of Borel sets is less than or equal to

In fact, the cardinality of the collection of Borel sets is equal to that of the continuum (compare to the number of Lebesgue measurable sets that exist, which is strictly larger and equal to ).

Standard Borel spaces and Kuratowski theorems

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See also: Standard Borel space

Let be a topological space. The Borel space associated to is the pair , where is the σ-algebra of Borel sets of .

George Mackey defined a Borel space somewhat differently, writing that it is "a set together with a distinguished σ-field of subsets called its Borel sets."[1] However, modern usage is to call the distinguished sub-algebra the measurable sets and such spaces measurable spaces. The reason for this distinction is that the Borel sets are the σ-algebra generated by open sets (of a topological space), whereas Mackey's definition refers to a set equipped with an arbitrary σ-algebra. There exist measurable spaces that are not Borel spaces, for any choice of topology on the underlying space.[2]

Measurable spaces form a category in which the morphisms are measurable functions between measurable spaces. A function is measurable if it pulls back measurable sets, i.e., for all measurable sets in , the set is measurable in .

Theorem. Let be a Polish space, that is, a topological space such that there is a metric on that defines the topology of and that makes a complete separable metric space. Then as a Borel space is isomorphic to one of

  1. ,
  2. ,
  3. a finite space.

(This result is reminiscent of Maharam's theorem.)

Considered as Borel spaces, the real line , the union of with a countable set, and are isomorphic.

A standard Borel space is the Borel space associated to a Polish space. A standard Borel space is characterized up to isomorphism by its cardinality,[3] and any uncountable standard Borel space has the cardinality of the continuum.

For subsets of Polish spaces, Borel sets can be characterized as those sets that are the ranges of continuous injective maps defined on Polish spaces. Note however, that the range of a continuous noninjective map may fail to be Borel. See analytic set.

Every probability measure on a standard Borel space turns it into a standard probability space.

Non-Borel sets

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An example of a subset of the reals that is non-Borel, due to Lusin,[4] is described below. In contrast, an example of a non-measurable set cannot be exhibited, although the existence of such a set is implied, for example, by the axiom of choice.

Every irrational number has a unique representation by an infinite simple continued fraction

where is some integer and all the other numbers are positive integers. Let be the set of all irrational numbers that correspond to sequences with the following property: there exists an infinite subsequence such that each element is a divisor of the next element. This set is not Borel. However, it is analytic (all Borel sets are also analytic), and complete in the class of analytic sets. For more details see descriptive set theory and the book by A. S. Kechris (see References), especially Exercise (27.2) on page 209, Definition (22.9) on page 169, Exercise (3.4)(ii) on page 14, and on page 196.

It's important to note, that while Zermelo–Fraenkel axioms (ZF) are sufficient to formalize the construction of , it cannot be proven in ZF alone that is non-Borel. In fact, it is consistent with ZF that is a countable union of countable sets,[5] so that any subset of is a Borel set.

Another non-Borel set is an inverse image of an infinite parity function . However, this is a proof of existence (via the axiom of choice), not an explicit example.

Alternative non-equivalent definitions

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According to Paul Halmos,[6] a subset of a locally compact Hausdorff topological space is called a Borel set if it belongs to the smallest σ-ring containing all compact sets.

Norberg and Vervaat[7] redefine the Borel algebra of a topological space as the -algebra generated by its open subsets and its compact saturated subsets. This definition is well-suited for applications in the case where is not Hausdorff. It coincides with the usual definition if is second countable or if every compact saturated subset is closed (which is the case in particular if is Hausdorff).

See also

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Notes

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References

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Further reading

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Borel set

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In , particularly in measure theory and , a Borel set is an element of the Borel on a , defined as the smallest that contains all open sets of the space. This is generated by iteratively applying countable unions, countable intersections, and complements starting from the open sets, resulting in a of sets including open sets (denoted Σ10\mathbf{\Sigma}_1^0), closed sets (Π10\mathbf{\Pi}_1^0), FσF_\sigma sets (countable unions of closed sets), GδG_\delta sets (countable intersections of open sets), and more complex levels up to the full Borel . 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 Rn\mathbb{R}^n. Borel sets play a central role in and , 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. The , a of modern integration , is defined precisely on the Borel σ-algebra of Rn\mathbb{R}^n, extending naturally to the completion that includes all Lebesgue measurable sets, though not all Lebesgue measurable sets are Borel (there are 22ℵ02^{2^{\aleph_0}} Lebesgue measurable sets but only continuum many Borel sets). 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 R\mathbb{R}. 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 GδG_\delta sets.

Generating the Borel σ-algebra

Definition and construction

In a topological space (X,τ)(X, \tau), where τ\tau denotes the collection of open sets, a σ\sigma-algebra on XX is a family of subsets of XX that contains the empty set and XX itself, and is closed under complementation relative to XX and countable unions (hence also countable intersections). The Borel σ\sigma-algebra B(X)\mathcal{B}(X) (or B(X)B(X)) is defined as the smallest σ\sigma-algebra on XX containing all open sets in τ\tau, or equivalently, the σ\sigma-algebra generated by τ\tau. Since the complement of an open set is closed, B(X)\mathcal{B}(X) is also generated by the closed sets of XX. The Borel sets, which form B(X)\mathcal{B}(X), consist of all subsets of XX that can be obtained from the open sets through countable applications of unions, intersections, and complements. This construction proceeds via transfinite over the ordinals up to the first uncountable ordinal ω1\omega_1. The process stratifies the Borel sets into a of classes Σα0\mathbf{\Sigma}^0_\alpha and Πα0\mathbf{\Pi}^0_\alpha for each countable ordinal α<ω1\alpha < \omega_1. The hierarchy begins with the base level: Σ10(X)\mathbf{\Sigma}^0_1(X) comprises the open sets of XX, while Π10(X)\mathbf{\Pi}^0_1(X) consists of the closed sets (complements of open sets). For a successor ordinal α+1<ω1\alpha + 1 < \omega_1, Σα+10(X)\mathbf{\Sigma}^0_{\alpha+1}(X) is the collection of all countable unions of sets from ⋃β<α+1Πβ0(X)\bigcup_{\beta < \alpha+1} \mathbf{\Pi}^0_\beta(X), and Πα+10(X)\mathbf{\Pi}^0_{\alpha+1}(X) is the collection of all countable intersections of sets from ⋃β<α+1Σβ0(X)\bigcup_{\beta < \alpha+1} \mathbf{\Sigma}^0_\beta(X). At a limit ordinal λ<ω1\lambda < \omega_1, the classes are defined as Σλ0(X)=⋃β<λΣβ0(X)\mathbf{\Sigma}^0_\lambda(X) = \bigcup_{\beta < \lambda} \mathbf{\Sigma}^0_\beta(X) and Πλ0(X)=⋃β<λΠβ0(X)\mathbf{\Pi}^0_\lambda(X) = \bigcup_{\beta < \lambda} \mathbf{\Pi}^0_\beta(X). The full Borel σ\sigma-algebra is then the union B(X)=⋃α<ω1Σα0(X)=⋃α<ω1Πα0(X)\mathcal{B}(X) = \bigcup_{\alpha < \omega_1} \mathbf{\Sigma}^0_\alpha(X) = \bigcup_{\alpha < \omega_1} \mathbf{\Pi}^0_\alpha(X).

Example in Euclidean space

In R\mathbb{R}, the standard topology is generated by the collection of all open intervals (a,b)(a, b) where a<ba < b, and any open set is a countable union of such disjoint open intervals. The Borel σ\sigma-algebra B(R)\mathcal{B}(\mathbb{R}) is then the smallest σ\sigma-algebra containing these open sets, so all open sets belong to B(R)\mathcal{B}(\mathbb{R}) by definition. Closed sets in R\mathbb{R} are complements of open sets and thus also Borel, forming the class Π10\Pi_1^0 in the Borel hierarchy. For instance, singletons {x}\{x\} for x∈Rx \in \mathbb{R} are closed (as complements of the open set R∖{x}\mathbb{R} \setminus \{x\}) and hence Borel. The middle-thirds Cantor set C⊂[0,1]C \subset [0,1], constructed by iteratively removing middle open intervals from [0,1][0,1], is a closed set (as the intersection of a decreasing sequence of closed sets) and therefore Borel with hierarchy level Π10\Pi_1^0. Countable unions of closed sets, known as FσF_\sigma sets or Σ20\Sigma_2^0 sets, are also Borel. The set of rational numbers Q\mathbb{Q} exemplifies this: enumerate Q={qn:n∈N}\mathbb{Q} = \{q_n : n \in \mathbb{N}\}, so Q=⋃n=1∞{qn}\mathbb{Q} = \bigcup_{n=1}^\infty \{q_n\}, a countable union of closed singletons, making Q\mathbb{Q} Borel at level Σ20\Sigma_2^0. Countable intersections of open sets, or GδG_\delta sets at level Π20\Pi_2^0, belong to the Borel σ\sigma-algebra as well. The irrationals R∖Q\mathbb{R} \setminus \mathbb{Q} form such a set: for an enumeration Q={qn:n∈N}\mathbb{Q} = \{q_n : n \in \mathbb{N}\}, R∖Q=⋂n=1∞(R∖{qn})\mathbb{R} \setminus \mathbb{Q} = \bigcap_{n=1}^\infty (\mathbb{R} \setminus \{q_n\}), a countable intersection of open sets, confirming its Borel status. In Rn\mathbb{R}^n equipped with the , open sets are countable unions of products of open intervals, such as ∏i=1n(ai,bi)\prod_{i=1}^n (a_i, b_i), which generate the Borel σ\sigma-algebra B(Rn)\mathcal{B}(\mathbb{R}^n). This Borel σ\sigma-algebra serves as the foundation for the Lebesgue σ\sigma-algebra, obtained by completing B(Rn)\mathcal{B}(\mathbb{R}^n) with respect to , where every Lebesgue measurable set differs from a Borel set by a set of measure zero.

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 ω1\omega_1. This hierarchy is fundamental in descriptive set theory, distinguishing sets based on the minimal ordinal level required to generate them from open sets. The hierarchy is denoted using boldface letters: for each countable ordinal α<ω1\alpha < \omega_1, the classes Σα0\boldsymbol{\Sigma}^0_\alpha, Πα0\boldsymbol{\Pi}^0_\alpha, and Δα0\boldsymbol{\Delta}^0_\alpha are defined recursively. The base levels are Σ10\boldsymbol{\Sigma}^0_1, the class of open sets, and Π10\boldsymbol{\Pi}^0_1, the class of closed sets, with Δ10=Σ10∩Π10\boldsymbol{\Delta}^0_1 = \boldsymbol{\Sigma}^0_1 \cap \boldsymbol{\Pi}^0_1 consisting of clopen sets. For successor ordinals α=β+1\alpha = \beta + 1, Σα0\boldsymbol{\Sigma}^0_\alpha consists of all countable unions of sets from Πβ0\boldsymbol{\Pi}^0_\beta, while Πα0\boldsymbol{\Pi}^0_\alpha consists of all countable intersections of sets from Σβ0\boldsymbol{\Sigma}^0_\beta (equivalently, complements of sets in Σα0\boldsymbol{\Sigma}^0_\alpha); the diagonal class is Δα0=Σα0∩Πα0\boldsymbol{\Delta}^0_\alpha = \boldsymbol{\Sigma}^0_\alpha \cap \boldsymbol{\Pi}^0_\alpha. At limit ordinals λ\lambda, the classes are defined as Σλ0=⋃α<λΣα0\boldsymbol{\Sigma}^0_\lambda = \bigcup_{\alpha < \lambda} \boldsymbol{\Sigma}^0_\alpha and Πλ0=⋃α<λΠα0\boldsymbol{\Pi}^0_\lambda = \bigcup_{\alpha < \lambda} \boldsymbol{\Pi}^0_\alpha. Early levels include Σ20\boldsymbol{\Sigma}^0_2, the FσF_\sigma sets (countable unions of closed sets), and Π20\boldsymbol{\Pi}^0_2, the GδG_\delta sets (countable intersections of open sets). The hierarchy is complete in the sense that the Borel σ-algebra is precisely the union ⋃α<ω1Σα0=⋃α<ω1Πα0=⋃α<ω1Δα0\bigcup_{\alpha < \omega_1} \boldsymbol{\Sigma}^0_\alpha = \bigcup_{\alpha < \omega_1} \boldsymbol{\Pi}^0_\alpha = \bigcup_{\alpha < \omega_1} \boldsymbol{\Delta}^0_\alpha. In uncountable Polish spaces, the hierarchy is strict: for each α<ω1\alpha < \omega_1, Σα0⊊Δα+10⊊Σα+10\boldsymbol{\Sigma}^0_\alpha \subsetneq \boldsymbol{\Delta}^0_{\alpha+1} \subsetneq \boldsymbol{\Sigma}^0_{\alpha+1} and similarly for the Π\boldsymbol{\Pi} classes, with no collapsing of levels. A key property of Borel sets within this hierarchy is captured by the perfect set theorem: every uncountable Borel set in a contains a perfect subset, which is closed, has no isolated points, and is homeomorphic to the . This theorem implies that Borel sets are either countable or have , highlighting their regularity compared to more complex descriptive classes.

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. 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. 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. On the real line R\mathbb{R}, the Borel σ-algebra B(R)\mathcal{B}(\mathbb{R}) has cardinality ∣B(R)∣=c|\mathcal{B}(\mathbb{R})| = \mathfrak{c}, the cardinality of the continuum, even though the power set of R\mathbb{R} has cardinality 2c2^{\mathfrak{c}}; this follows from the Borel hierarchy, where open sets number c\mathfrak{c} many, and transfinite iterations of countable unions, intersections, and complements up to length ω1\omega_1 yield at most cℵ1=c\mathfrak{c}^{\aleph_1} = \mathfrak{c} sets overall. In metric spaces such as Rn\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,B(X))(X, \mathcal{B}(X)), where XX is a Polish space—that is, a separable completely metrizable topological space—and B(X)\mathcal{B}(X) denotes the Borel σ\sigma-algebra on XX generated by its open sets. More generally, any measurable space Borel isomorphic to such a pair (X,B(X))(X, \mathcal{B}(X)) is also called a standard Borel space. Standard Borel spaces are characterized by having a countably generated σ\sigma-algebra that separates points. A Borel isomorphism between measurable spaces (X,BX)(X, \mathcal{B}_X) and (Y,BY)(Y, \mathcal{B}_Y) is a f:X→Yf: X \to Y such that ff and its inverse f−1f^{-1} are both measurable functions, meaning they map Borel sets in their respective σ\sigma-algebras to Borel sets. This preserves the Borel structure, allowing standard Borel spaces to be characterized up to their . Prominent examples of standard Borel spaces include the Euclidean spaces Rn\mathbb{R}^n equipped with the standard , the 2N2^\mathbb{N} consisting of infinite binary sequences with the , and the separable ℓ2\ell^2 of square-summable real sequences with its norm . 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 . Uncountable standard Borel spaces exhibit an atomless structure in their σ\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 class, positions standard Borel spaces as the setting for classical descriptive , where Borel sets and their hierarchies are analyzed systematically.

Kuratowski theorems

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

Non-Borel Sets

Existence and construction

The existence of non-Borel sets follows from a cardinality argument. The power set P(R)\mathcal{P}(\mathbb{R}) of the real numbers has cardinality 2c2^{\mathfrak{c}}, where c=∣R∣\mathfrak{c} = |\mathbb{R}| denotes the , while the Borel σ\sigma-algebra B(R)\mathcal{B}(\mathbb{R}) has cardinality c\mathfrak{c}. Consequently, there are 2c2^{\mathfrak{c}} many subsets of R\mathbb{R} that are not Borel sets. Explicit constructions of non-Borel sets rely on the (AC). One such example is the , introduced by Giuseppe Vitali in 1905. Consider the on R\mathbb{R} defined by x∼yx \sim y if and only if x−y∈Qx - y \in \mathbb{Q}. The equivalence classes are the s x+Qx + \mathbb{Q}. Using AC, select one representative from each intersected with the interval [0,1)[0,1), yielding a set V⊂[0,1)V \subset [0,1). The set VV is non-Lebesgue measurable, as its countable rational translates cover $[0,1]withoutoverlapbutleadtoacontradictionundertheadditivityof[Lebesguemeasure](/page/Lebesguemeasure).SinceallBorelsetsareLebesguemeasurable,without overlap but lead to a contradiction under the additivity of [Lebesgue measure](/page/Lebesgue_measure). Since all Borel sets are Lebesgue measurable,V$ cannot be Borel. Another construction, due to Felix Bernstein in 1905, yields a Bernstein set B⊂RB \subset \mathbb{R}, which intersects every uncountable closed subset of R\mathbb{R} (every ), as does its complement R∖B\mathbb{R} \setminus B. To build BB, enumerate all perfect subsets of R\mathbb{R} as {Pα:α<c}\{P_\alpha : \alpha < \mathfrak{c}\} via a well-ordering of type c\mathfrak{c}. Proceed by transfinite : at stage α\alpha, select two distinct points pα,qα∈Pα∖⋃β<α{pβ,qβ}p_\alpha, q_\alpha \in P_\alpha \setminus \bigcup_{\beta < \alpha} \{p_\beta, q_\beta\} using AC, assigning pαp_\alpha to BB and qαq_\alpha to R∖B\mathbb{R} \setminus B. The resulting BB lacks the Baire property, meaning it is neither meager nor comeager in any non-empty . As all Borel sets possess the Baire property, BB is non-Borel. The Sierpiński set, introduced by in 1924, provides another example under the . It is an uncountable subset S⊂RS \subset \mathbb{R} such that S∩NS \cap N is at most countable for every Lebesgue N⊂RN \subset \mathbb{R}. The existence of such sets is independent of ZFC; the construction employs AC and transfinite recursion by enumerating all null GδG_\delta sets (of which there are c\mathfrak{c} many) and selecting points at each stage from the complement of the union of previously enumerated null GδG_\delta sets, ensuring the intersection property. Such a set SS is non-Lebesgue measurable, hence non-Borel.

Implications for measure theory

The Lebesgue σ-algebra on R\mathbb{R} is defined as the completion of the with respect to , incorporating all along with all subsets of Borel null sets and their complements. This structure ensures that every is Lebesgue measurable, but the Lebesgue σ-algebra is strictly larger, with exceeding that of the continuum, allowing certain non-Borel sets to be Lebesgue measurable precisely when they differ from by null sets. The 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. 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 R\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][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 R\mathbb{R} cannot admit a complete translation-invariant measure, necessitating the restricted domain of the Lebesgue σ-algebra for consistency. In , 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 for absolutely continuous cases. 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. Nonetheless, in advanced 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. 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 . The Lusin–Suslin theorem, established in 1917, guarantees that every analytic set is . Suslin's theorem, also from 1917, states that a set is Borel if and only if it is both analytic and coanalytic. These theorems resolve a key gap by confirming the measurability of these projective sets under , thereby extending the applicability of measure theory to broader classes without encountering non-measurable exceptions in this regime. 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.

Alternative Definitions

Equivalent characterizations

In a topological space XX, the Borel σ\sigma-algebra B(X)\mathcal{B}(X) can equivalently be defined as the smallest σ\sigma-algebra containing all closed subsets of XX. This follows from the fact that the closed sets are precisely the complements of the open sets, and any σ\sigma-algebra closed under complements and containing the opens must contain the closeds, and vice versa. Another equivalent characterization is that B(X)\mathcal{B}(X) is the smallest σ\sigma-algebra on XX such that every f:X→Rf: X \to \mathbb{R} is measurable with respect to B(X)\mathcal{B}(X) and the Borel σ\sigma-algebra on R\mathbb{R}. Indeed, the preimage under any continuous ff of an in R\mathbb{R} is open in XX, so lies in B(X)\mathcal{B}(X); conversely, the open sets in XX are generated as such preimages from continuous functions to R\mathbb{R}. In the case of a locally compact XX, B(X)\mathcal{B}(X) is also the σ\sigma-algebra generated by the compact subsets of XX. Here, every compact set is closed, and in such spaces, the compact sets form a basis for generating the via their complements and unions, yielding the same σ\sigma-algebra as the opens. 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×NNX \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.

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. 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. In descriptive set theory, the projective hierarchy begins with the analytic sets, denoted Σ¹₁, defined as the continuous images of from Polish spaces such as the ω^ω. This class properly contains the , 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 . The existence of non-Borel analytic sets was first demonstrated by Sierpiński in , highlighting how projection or continuous preimages can escape the while remaining "definable" in a broader sense. Higher levels of the projective hierarchy, like Π¹₁ coanalytic sets, further extend this structure. Early 20th-century proposals for classes resembling Borel sets included variants by , who in introduced a transfinite 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 of Borel sets explored how such generations yield sets of 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.

References

  1. https://mathworld.wolfram.com/BorelSet.html
  2. https://faculty.etsu.edu/gardnerr/5210/notes/1-4.pdf
  3. https://mathshistory.st-andrews.ac.uk/Biographies/Borel/
  4. https://www.math.utep.edu/faculty/nsanyal/blog/measure_and_prob_1.html
  5. https://home.mathematik.uni-freiburg.de/maxwell/coursenotes-descriptivesettheory-website.pdf
  6. https://www.math.stonybrook.edu/~aknapp/books/basic/b-ra-Ch6-sample.pdf
  7. https://assets.press.princeton.edu/chapters/s8008.pdf
  8. https://home.iitk.ac.in/~chavan/Analysis.pdf
  9. https://www.math.ucdavis.edu/~hunter/m206/ch1_measure.pdf
  10. http://homepages.math.uic.edu/~marker/math512/dst.pdf
  11. https://www.math.ucla.edu/~ynm/lectures/dst2009/dst2009.pdf
  12. https://link.springer.com/book/10.1007/978-1-4612-4190-4
  13. https://www.reed.edu/math/wieting/essays/AnalyticBorelSpaces.pdf
  14. https://link.springer.com/article/10.1007/s10474-019-00986-7
  15. https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/bachelor/2017-2018/barinaga-bsc-scriptie.pdf
  16. https://arxiv.org/pdf/1406.3062
  17. https://www.math.ucdavis.edu/~hunter/m201b_old/measure.pdf
  18. https://homepages.math.uic.edu/~marker/math512/dst.pdf
  19. https://www.colorado.edu/amath/sites/default/files/attached-files/billingsley.pdf
  20. https://books.google.com/books/about/Probability_and_Measure.html?id=BfBQAAAAMAAJ
  21. https://www2.stat.duke.edu/~sayan/CBB2012/1-1-Algebraandsigma.pdf
  22. https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch3.pdf
  23. http://www.math.brown.edu/~treil/teaching/MA221-f03/Notes/Sept_8/Sept_8.pdf
  24. https://people.clas.ufl.edu/mjury/files/6616notes18nov2015.pdf
  25. https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-15/issue-1/On-measure-compactness-and-Borel-measure-compactness/ojm/1200770910.pdf
  26. https://www.ams.org/bookstore/pspdf/surv-155-intro.pdf
  27. https://almostsuremath.com/2018/12/24/analytic-sets/
  28. https://math.stackexchange.com/questions/155261/baire-sigma-algebra
  29. https://heil.math.gatech.edu/6337/spring11/section2.4.pdf
  30. https://www.math3ma.com/blog/lebesgue-but-not-borel
  31. https://math.berkeley.edu/~slaman/papers/sierpinski.pdf
  32. https://topology.lmf.cnrs.fr/borel-sets-analytic-sets-and-the-baire-property/
  33. https://www.math.uni-bonn.de/people/koepke/Preprints/The_influence_of_Felix_Hausdorff_on_the_early_development_of_descriptive_set_theory.pdf
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