Borel σ-algebra

The Borel σ-algebra is a set system in measure theory and essential for the axiomatic structure of modern stochastics and integration theory . Borel's σ-algebra is a σ-algebra that contains all sets to which one naively wants to assign a volume or a probability, but excludes negative results such as Vitali's theorem .

Borel's σ-algebra is particularly important because it is naturally adapted to the structure of topological spaces and thus to both metric and standardized spaces . This can be seen, among other things, in the fact that all continuous functions are always measurable with respect to Borel's σ-algebra .

The sets contained in Borel's σ-algebra can only be fully described in very rare cases. Conversely, however, it is difficult to construct a set that does not lie in Borel's σ-algebra. As a rough rule of thumb, it should contain “almost all of the amounts that occur” or “every amount that can be constructed constructively”.

The sets contained in the Borel σ-algebra are called Borel sets , Borel sets or also Borel measurable sets . The naming of the σ-algebra and the quantities follows in honor of Émile Borel , who implicitly used them for the first time in 1898.

definition

A topological space is given , where the set system is open sets . ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle {\mathcal {O}}}$

Then the σ-algebra generated by is called the Borelian σ-algebra. It is referred to as or, if the amount is obvious from the context, also as . ${\displaystyle {\mathcal {O}}}$ ${\displaystyle {\mathcal {B}}(X)}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {B}}}$

So it is

${\displaystyle {\mathcal {B}}(X):=\sigma ({\mathcal {O}})}$,

where here denotes the σ operator . The Borel σ-algebra is thus defined as the smallest σ-algebra (with regard to set inclusion) that contains all open sets. ${\displaystyle \sigma (\cdot )}$

Remarks

• The Borel σ-algebra is always uniquely determined.
• A Borel σ-algebra thus makes it possible to equip a topological space canonically with the additional structure of a measurement space. With regard to this structure, the room is then also called Borel room. However, other measuring rooms are also referred to as Borel rooms.
• For metric spaces and standardized spaces , the topology generated by the metric or norm is selected as the topology.
• The sets contained in Borel's σ-algebra are called Borel sets. The class of Borel sets is a subclass of the class of Suslin or analytic sets .

The Borel σ-algebra on the real numbers

The set of real numbers is usually equipped with the topology that is spanned by the open intervals with rational end points. This makes Borel's σ-algebra a separable σ-algebra . Although other topologies on are also considered in individual cases , this is considered the canonical topology on , and the Borelian σ-algebra derived from it is simply referred to as the Borelian σ-algebra on . ${\displaystyle \mathbb {R} }$ ${\displaystyle (a,b)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$

The Borel σ-algebra of does not contain all subsets of . It can even be shown that the Borel σ-algebra of is of equal power to , while the set of all subsets of has a really greater power than . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$

Producer

The Borel σ-algebra is not defined directly, but implicitly via a generator. This is a given system of sets that generates Borel's σ-algebra in the sense that it is the smallest σ-algebra that preserves all the sets of the generator. For details see Creator of a σ-Algebra . Some of the possible producers are the following: ${\displaystyle {\mathcal {E}}}$

• ${\displaystyle {\mathcal {E}}_{0}=\{A\subset \mathbb {R} \mid A{\text{ ist offen }}\}}$ by definition
• ${\displaystyle {\mathcal {E}}_{1}=\{[a,b]\subset \mathbb {R} \mid a\leq b\}}$ or ${\displaystyle {\mathcal {E}}_{2}=\{[a,b]\subset \mathbb {R} \mid a\leq b,\;a,b\in \mathbb {Q} \}}$
• ${\displaystyle {\mathcal {E}}_{3}=\{(a,b)\subset \mathbb {R} \mid a<b\}}$ or ${\displaystyle {\mathcal {E}}_{4}=\{(a,b)\subset \mathbb {R} \mid a<b,\;a,b\in \mathbb {Q} \}}$
• ${\displaystyle {\mathcal {E}}_{5}=\{(a,b]\subset \mathbb {R} \mid a\leq b\}}$ or ${\displaystyle {\mathcal {E}}_{6}=\{(a,b]\subset \mathbb {R} \mid a\leq b,\;a,b\in \mathbb {Q} \}}$
• ${\displaystyle {\mathcal {E}}_{7}=\{(-\infty ,a]\mid a\in \mathbb {R} \}}$ or ${\displaystyle {\mathcal {E}}_{8}=\{(-\infty ,a]\mid a\in \mathbb {Q} \}}$
• ${\displaystyle {\mathcal {E}}_{9}=\{(-\infty ,a)\mid a\in \mathbb {R} \}}$ or ${\displaystyle {\mathcal {E}}_{10}=\{(-\infty ,a)\mid a\in \mathbb {Q} \}}$

In particular, there are obviously several generators for the Borel σ-algebra. However, the Borel σ-algebra is uniquely determined by specifying a generator. The choice of the specific producer often depends on the situation. Often one chooses quantity systems that are stable on the average as the producer, since with them, according to the principle of uniqueness of dimensions, a measure is already clearly determined by the specification of the values ​​on the producer. When using distribution functions, producers to offer themselves . The intervals with rational limits are often used for approximation arguments. In particular, the generators and half rings listed here (if one defines each so that the generators contain the empty set). ${\displaystyle {\mathcal {E}}_{7}}$ ${\displaystyle {\mathcal {E}}_{10}}$ ${\displaystyle {\mathcal {E}}_{5}}$ ${\displaystyle {\mathcal {E}}_{6}}$ ${\displaystyle (a,a]:=\emptyset }$

Amounts included

The sets contained in Borel's σ-algebra are abundant. It contains

• all open sets , all closed sets, and all compact sets
• all intervals of the form for as well as and ${\displaystyle (a,b),[a,b],(a,b],[a,b)}$ ${\displaystyle a,b\in \mathbb {R} }$ ${\displaystyle (-\infty ,b),(-\infty ,b],(a,\infty )}$ ${\displaystyle [a,\infty )}$
• all point sets, i.e. sets of the form for and all finite subsets of and all countably infinite subsets of ${\displaystyle \{a\}}$ ${\displaystyle a\in \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$
• From the defining properties of σ-algebras it follows directly that finite and countably infinite unions and cuts of Borel sets are again Borel sets, as are the difference and the complement.
• If continuous, then archetypes of Borel sets are again Borel sets, in particular also level sets , sub- level sets , and super level sets . ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$

The Borel σ-algebra on the extended real numbers

Sometimes the real numbers are extended by the values , one then names accordingly ${\displaystyle \pm \infty }$

${\displaystyle {\overline {\mathbb {R} }}:=\mathbb {R} \cup \{+\infty ,-\infty \}}$

the extended real numbers . They occur, for example, when examining numerical functions . The Borel σ-algebra on the extended real numbers is then explained by

${\displaystyle {\mathcal {B}}({\overline {\mathbb {R} }}):=\{A\cup E\mid A\in {\mathcal {B}}(\mathbb {R} ),\;E\subseteq \{-\infty ,\infty \}\}}$.

It therefore consists of all Borel sets on the real numbers and of these Borel sets combined with , or . ${\displaystyle \{\infty \}}$ ${\displaystyle \{-\infty \}}$ ${\displaystyle \{\infty ,-\infty \}}$

Further Borel σ-algebras

The Borel σ-algebra on separable metric spaces

A separable metric space is given . The open spheres create a topology as a basis, this is called the topology created by the metric. Because of the separability (which in the metric case is equivalent to the second axiom of countability ) every open set is to be written as a countable union of open spheres. The smallest algebra that contains the open spheres therefore contains all open sets and is therefore equal to Borel's algebra. ${\displaystyle (X,d)}$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$

The special case and the Euclidean metric are discussed in more detail in the following sections. ${\displaystyle X\subseteq \mathbb {R} ^{n}}$ ${\displaystyle d}$

The Borel σ-algebra on finite-dimensional real vector spaces

The canonical topology of the -dimensional cuboids with rational coordinates and is spanned on the finite-dimensional vector spaces . It is also the -fold product topology of the canonical topology . The Borel of it generated σ-algebra is analogous to the one-dimensional case the Borel σ-algebra on . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle n}$ ${\displaystyle (a_{1},b_{1})\times \dotsb \times (a_{n},b_{n})}$ ${\displaystyle a_{i}}$ ${\displaystyle b_{i}}$ ${\displaystyle n}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} ^{n}}$

In this way the Borel σ-algebra of complex numbers is elegantly explained: One simply uses the vector space isomorphism between and . ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} ^{2}}$

Subsets that do not belong to the Borel σ-algebra usually have an intuitively exotic character. In three-dimensional real space, the sets that are used in the Banach-Tarski paradox are an example of such subsets that do not belong to Borel's σ-algebra.

Borel's σ-algebra on general topological spaces

The properties of Borel's σ-algebra in any topological space depend essentially on the structure of the topological space. In general it can only be said that Borel's σ-algebra always contains all open (by definition) and all closed sets (due to the complementary stability).

The more structure the topological space has, the more sets the Borel σ-algebra then also contains. The following applies:

• If the topological space is a T1 space , then all one-element sets are contained in Borel's σ-algebra. Thus, all finite sets, all countably infinite sets and all sets with finite or countably infinite complement are contained in Borel's σ-algebra.
• If the topological space is a Hausdorff space (such as a metric space ), then all compact sets are closed and thus contained in Borel's σ-algebra.

Product spaces and the Borel σ-algebra

If two topological spaces and are given, the Borel σ-algebra can be defined in two ways: ${\displaystyle (X_{1},{\mathcal {O}}_{1})}$ ${\displaystyle (X_{2},{\mathcal {O}}_{2})}$

• Either one forms the (topological) product space , provided with the product topology , here denoted by. The Borel σ-algebra on can then be defined as the Borel σ-algebra of the product topology, i.e. as ${\displaystyle X_{1}\times X_{2}}$ ${\displaystyle {\mathcal {O}}_{1}\otimes {\mathcal {O}}_{2}}$ ${\displaystyle X_{1}\times X_{2}}$

${\displaystyle {\mathcal {B}}(X_{1}\times X_{2}):=\sigma ({\mathcal {O}}_{1}\otimes {\mathcal {O}}_{2})}$

• or one first forms the Borel σ-algebras of the individual topological spaces and then their product σ-algebra , here denoted by: ${\displaystyle \otimes }$

${\displaystyle {\mathcal {B}}(X_{1}\times X_{2}):=\sigma ({\mathcal {O}}_{1})\otimes \sigma ({\mathcal {O}}_{2})}$

In fact, the two constructions agree in many cases, even if the question is extended to families of topological spaces . The following applies: ${\displaystyle (X_{i})_{i\in I}}$

If is a countable family of topological spaces, each of which has a countable basis ( i.e. satisfies the second axiom of countability ), and be the topological product of all these spaces, then is ${\displaystyle (X_{i})_{i\in I}}$ ${\displaystyle X}$

${\displaystyle {\mathcal {B}}(X)=\bigotimes _{i\in I}{\mathcal {B}}(X_{i})}$.

The Borelian σ-algebra of the product is therefore the product σ-algebra of the Borelian σ-algebra. The statement applies in particular to all separable metric spaces and thus also to . So is ${\displaystyle \mathbb {R} }$

${\displaystyle {\mathcal {B}}(\mathbb {R} ^{2})={\mathcal {B}}(\mathbb {R} )\otimes {\mathcal {B}}(\mathbb {R} )}$.

Nomenclature for certain Borel quantities

→ Main article : Borel hierarchy

• In the literature, the following designation, introduced by Felix Hausdorff , has established itself for some simple classes of Borel sets:

- with are all unions of countably many closed sets, ${\displaystyle \operatorname {F} _{\sigma }}$

- with all averages of countably many open sets, ${\displaystyle \operatorname {G} _{\delta }}$

- with all averages of countably many quantities, ${\displaystyle \operatorname {F} _{\sigma \delta }}$ ${\displaystyle \operatorname {F} _{\sigma }}$

- with all unions of countably many sets, ${\displaystyle \operatorname {G} _{\delta \sigma }}$ ${\displaystyle \operatorname {G} _{\delta }}$

- with all unions of countably many sets, ${\displaystyle \operatorname {F} _{\sigma \delta \sigma }}$ ${\displaystyle \operatorname {F} _{\sigma \delta }}$

- with all averages of countably many amounts ${\displaystyle \operatorname {G} _{\delta \sigma \delta }}$ ${\displaystyle \operatorname {G} _{\delta \sigma }}$

etc.

All , , , , , , ...- sets are Borel sets. However, this scheme does not make it possible to describe all Borel sets, because the union of all these classes is not yet completed in general with regard to the axioms of an -algebra. ${\displaystyle \operatorname {F} _{\sigma }}$ ${\displaystyle \operatorname {G} _{\delta }}$ ${\displaystyle \operatorname {F} _{\sigma \delta }}$ ${\displaystyle \operatorname {G} _{\delta \sigma }}$ ${\displaystyle \operatorname {F} _{\sigma \delta \sigma }}$ ${\displaystyle \operatorname {G} _{\delta \sigma \delta }}$ ${\displaystyle \sigma }$

• In descriptive set theory , the open sets are also referred to as -sets, the -sets as -sets, the -sets as -sets, etc. are called complements of -sets -sets; for example the quantities are exactly the quantities. ${\displaystyle \Sigma _{1}^{0}}$ ${\displaystyle \operatorname {F} _{\sigma }}$ ${\displaystyle \Sigma _{2}^{0}}$ ${\displaystyle \operatorname {G} _{\delta \sigma }}$ ${\displaystyle \Sigma _{3}^{0}}$ ${\displaystyle \Sigma _{n}^{0}}$ ${\displaystyle \Pi _{n}^{0}}$ ${\displaystyle \Pi _{2}^{0}}$ ${\displaystyle \operatorname {G} _{\delta }}$

application

The set together with Borel's σ-algebra is a measurement space and forms the basis of the Borel measures as such. All elements of Borel's σ-algebra (which are themselves sets) are called Borel-measurable; only these are assigned values ​​by means of a Borel measure. ${\displaystyle \Omega }$

Individual evidence

1. ^ Georgii: Stochastics. 2009, p. 12.
2. ↑ Klenke: Probability Theory. 2013, p. 8.
3. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 17.
4. ↑ ^{a b} Pavel S. Alexandroff: Textbook of set theory. 6th, revised edition. Harri Deutsch, Thun et al. 1994, ISBN 3-8171-1365-X .
5. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 115.
6. ↑ Vladimir Kanovei, Peter Koepke: Descriptive set theory in Hausdorff's basics of set theory. 2001, uni-bonn.de (pdf; 267 kB) .
7. ↑ Isidor P. Natanson: Theory of the functions of a real variable. Unchanged reprint of the 4th edition. Harri Deutsch, Thun et al. 1977, ISBN 3-87144-217-8 (also in digital form in Russian at INSTITUTE OF COMPUTATIONAL MODELING SB RAS, Krasnoyarsk ).
8. ↑ At z. B. it is only possible with the help of transfinite ordinal numbers to continue this system in such a way that all Borel sets are recorded by it (see Baie classes: connection to the Borel sets ). But there are also topological spaces in which the - and - sets alone exhaust the whole class of Borel sets, e.g. B. in a T ₁ space with a countable number of points. More on this topic can be found in Felix Hausdorff : Set theory. 2nd, revised edition. de Gruyter, Berlin et al. 1927, can be read. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \operatorname {F} _{\sigma }}$ ${\displaystyle \operatorname {G} _{\delta }}$

literature

• Sashi M. Srivastava: A course on Borels sets (= Graduate Texts in Mathematics. Vol. 180). Springer, New York NY et al. 1998, ISBN 0-387-98412-7 .
• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .
• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .