Borel set

 ( Topology ) 

In mathematics, the Borel sets included in a topological space are a particular class of "well-behaved" subsets of that space. For example, whereas an arbitrary subset of the 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 $X$ that contains both the empty set both the entire set $X$ and is closed under countable set union and countable intersection.

Then we can define the Borel σ-algebra over $X$ to be the smallest σ-algebra containing all open sets of $X$. A Borel subset of $X$ 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 of the topological space, rather than the open sets. The two definitions are equivalent for many well-behaved spaces, including all Hausdorff space σ-compact spaces, but can be different in more pathological spaces.

---

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

For a collection $T$ of subsets of $X$ (that is, for any subset of the power set $\backslash mathcal\; P$$(X)$ of $X$), let

• $T\_\backslash sigma$ be all countable unions of elements of $T$
• $T\_\backslash delta$ be all countable intersections of elements of $T$
• $T\_\{\backslash delta\backslash sigma\}\; =\; (T\_\backslash delta)\_\backslash sigma.$

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

• For the base case of the definition, let $G^0$ be the collection of open subsets of $X$.
• If $i$ is not a limit ordinal, then $i$ has an immediately preceding ordinal $i-1$. Let $$G^i\; =\; G^\{i-1\}\_\{\backslash delta\; \backslash sigma\}.$$
• If $i$ is a limit ordinal, set

The claim is that the Borel algebra is $G^\{\backslash omega\_1\}$, where $\{\backslash omega\_1\}$ is the first uncountable ordinal number. That is, the Borel algebra can be generated from the class of open sets by iterating the operation $$G\; \backslash mapsto\; G\_\{\backslash delta\; \backslash sigma\}$$ 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 $G^m$ into itself for any limit ordinal $m$; moreover if $m$ is an uncountable limit ordinal, $G^m$ is closed under countable unions.

For each Borel set $B$, there is some countable ordinal $\backslash alpha\_B$ such that $B$ can be obtained by iterating the operation over $\backslash alpha\_B$. However, as $B$ varies over all Borel sets, $\backslash alpha\_B$ will vary over all the countable ordinals, and thus the first ordinal at which all the Borel sets are obtained is $\backslash omega\_1$, the first uncountable ordinal.

The resulting sequence of sets is termed the Borel hierarchy.

---

Example An important example, especially in the theory of probability, is the Borel algebra on the set of . 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 $\backslash mathbb\; R$ that contains all the intervals.

In the construction by transfinite induction, it can be shown that, in each step, the cardinality of sets is, at most, the cardinality of the continuum. So, the total number of Borel sets is less than or equal to $$\backslash aleph\_1\; \backslash cdot\; 2\; ^\; \{\backslash aleph\_0\}\backslash ,\; =\; 2^\{\backslash aleph\_0\}.$$

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 $2^\{2^\{\backslash aleph\_0\}\}$).

---

Standard Borel spaces and Kuratowski theorems Let $X$ be a topological space. The Borel space associated to $X$ is the pair $(X,\backslash mathcal\; B)$, where $\backslash mathcal\; B$ is the σ-algebra of Borel sets of $X$.

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." However, modern usage is to call the distinguished sub-algebra the measurable sets and such spaces measurable space. 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. Jochen Wengenroth, Is every sigma-algebra the Borel algebra of a topology?

Measurable spaces form a category in which the are measurable functions between measurable spaces. A function $f:X\; \backslash rightarrow\; Y$ is measurable if it pullback measurable sets, i.e., for all measurable sets $B$ in $Y$, the set $f^\{-1\}(B)$ is measurable in $X$.

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

1. $\backslash mathbb\; R$,
2. $\backslash mathbb\; Z$,
3. a finite space.

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

Considered as Borel spaces, the real line $\backslash mathbb\; R$, the union of $\backslash mathbb\; R$ with a countable set, and $\backslash mathbb\; R^n$ 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, 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 An example of a subset of the reals that is non-Borel, due to Nikolai Luzin, 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

$x\; =\; a\_0\; +\; \backslash cfrac\{1\}\{a\_1\; +\; \backslash cfrac\{1\}\{a\_2\; +\; \backslash cfrac\{1\}\{a\_3\; +\; \backslash cfrac\{1\}\{\backslash ddots\backslash ,\}\}\}\}$

where $a\_0$ is some integer and all the other numbers $a\_k$ are positive integers. Let $A$ be the set of all irrational numbers that correspond to sequences $(a\_0,a\_1,\backslash dots)$ with the following property: there exists an infinite subsequence $(a\_\{k\_0\},a\_\{k\_1\},\backslash dots)$ such that each element is a divisor of the next element. This set $A$ is not Borel. However, it is analytic set (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 $A$, it cannot be proven in ZF alone that $A$ is non-Borel. In fact, it is consistent with ZF that $\backslash mathbb\{R\}$ is a countable union of countable sets, so that any subset of $\backslash mathbb\{R\}$ is a Borel set.

Another non-Borel set is an inverse image $f^\{-1\}0$ of an infinite parity function $f\backslash colon\; \backslash \{0,\; 1\backslash \}^\{\backslash omega\}\; \backslash to\; \backslash \{0,\; 1\backslash \}$. However, this is a proof of existence (via the axiom of choice), not an explicit example.

---

Alternative non-equivalent definitions According to Paul Halmos, a subset of a locally compact Hausdorff topological space is called a Borel set if it belongs to the smallest Sigma-ring containing all compact sets.

Norberg and VervaatTommy Norberg and Wim Vervaat, Capacities on non-Hausdorff spaces, in: Probability and Lattices, in: CWI Tract, vol. 110, Math. Centrum Centrum Wisk. Inform., Amsterdam, 1997, pp. 133-150 redefine the Borel algebra of a topological space $X$ as the $\backslash sigma$-algebra generated by its open subsets and its compact Saturated set. This definition is well-suited for applications in the case where $X$ is not Hausdorff. It coincides with the usual definition if $X$ is second countable or if every compact saturated subset is closed (which is the case in particular if $X$ is Hausdorff).

---

See also

---

Notes

---

Further reading

• William Arveson, An Invitation to C*-algebras, Springer-Verlag, 1981. (See Chapter 3 for an excellent exposition of Polish topology)
• Richard Dudley, Real Analysis and Probability. Wadsworth, Brooks and Cole, 1989
• See especially Sect. 51 "Borel sets and Baire sets".
• Halsey Royden, Real Analysis, Prentice Hall, 1988
• Alexander S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995 (Graduate texts in Math., vol. 156)

---

External links

• Formal definition of Borel Sets in the Mizar system, and the list of theorems that have been formally proved about it.

Post Comment