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\end{align*}
 
\end{align*}
 
In particular, a set of the form $ \{(x,y)\in\R^2:x-y\in B\} $ belongs to $ \A_{1,1}$ if and only if $B\in\B_1$.
 
In particular, a set of the form $ \{(x,y)\in\R^2:x-y\in B\} $ belongs to $ \A_{1,1}$ if and only if $B\in\B_1$.
 +
 +
====Relations to topological spaces====
  
 
Every topology generates a σ-algebra, called Borel σ-algebra. That is, the Borel σ-algebra on a [[topological space]] is, by definition, generated by the open sets.
 
Every topology generates a σ-algebra, called Borel σ-algebra. That is, the Borel σ-algebra on a [[topological space]] is, by definition, generated by the open sets.
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That is instructive: topological spaces are not a prerequisite to measurable spaces.
 
That is instructive: topological spaces are not a prerequisite to measurable spaces.
  
The Borel σ-algebra is not the only bridge between topological and measurable spaces. All sets having the [[Baire property]] (sometimes called Baire sets, which may be confusing) are a σ-algebra (generated by open sets together with  
+
A Borel measurable map is generally not continuous, and a Borel isomorphism is generally not a homeomorphism. However, every Borel measurable homomorphism between Polish groups is continuous. Accordingly, the topology of a Polish group is uniquely determined by its Borel σ-algebra (see [3, Sect. 9.C], [4, Sect. 1.2]).
[[Category of a set|meager]] sets) greater than Borel [3, Sect.8.F]. On the other hand, all compact [[Set of type F sigma(G delta)|$G_\delta$ subsets]] of a [[Compact space|compact]] [[Hausdorff space|Hausdorff]] topological space generate a σ-algebra (smaller than Borel) of sets called [[Baire set]]s in [4, Sect.7.1], [5, Sect.51]. For more general (in particular, uncountable [[Discrete space|discrete]]) topological spaces the definitions of [4] and [5] disagree. Note also the σ-algebra of [[Perfect measure|universally measurable]] sets.
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''Example.'' Consider again a separable Hilbert space $H$ and its Borel σ-algebra $\B$. A set $U\subset H$ contains a neighborhood of the origin if and only if there exist $A_1,A_2,\dots\in\B$ such that $A_1\cup A_2\cup\dots=H$ and $A_n-A_n\subset U$ for all $n$ (that is, $x-y\in U$ for all $x,y\in A_n$).
 +
 
 +
The Borel σ-algebra is not the only bridge between topological and measurable spaces. All sets having the [[Baire property]] (sometimes called Baire sets, which may be confusing) are a σ-algebra (generated by open sets together with [[Category of a set|meager]] sets) greater than Borel [3, Sect.8.F]. On the other hand, all compact [[Set of type F sigma(G delta)|$G_\delta$ subsets]] of a [[Compact space|compact]] [[Hausdorff space|Hausdorff]] topological space generate a σ-algebra (smaller than Borel) of sets called [[Baire set]]s in [5, Sect.7.1], [6, Sect.51]. For more general (in particular, uncountable [[Discrete space|discrete]]) topological spaces the definitions of [5] and [6] disagree. Note also the σ-algebra of [[Perfect measure|universally measurable]] sets.
  
 
====Older terminology====
 
====Older terminology====
  
Weaker assumptions on $\A$ were usual in the past. For example, according to [5], $\A$ need not contain the whole $X$, it is a [[Ring of sets|σ-ring]], not necessarily a σ-algebra. According to [6], a measurable space is not a pair $(X,\A)$ but a [[measure space]] $(X,\A,\mu)$ such that $X\in\A$ (and again, $\A$ is generally a σ-ring).
+
Weaker assumptions on $\A$ were usual in the past. For example, according to [6], $\A$ need not contain the whole $X$, it is a [[Ring of sets|σ-ring]], not necessarily a σ-algebra. According to [7], a measurable space is not a pair $(X,\A)$ but a [[measure space]] $(X,\A,\mu)$ such that $X\in\A$ (and again, $\A$ is generally a σ-ring).
  
 
====References====
 
====References====
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<TR><TD valign="top">[2]</TD> <TD valign="top">David Pollard, "A user's guide to measure theoretic probability", Cambridge (2002) |  {{MR|1873379}} | {{ZBL|0992.60001}}</TD></TR>
 
<TR><TD valign="top">[2]</TD> <TD valign="top">David Pollard, "A user's guide to measure theoretic probability", Cambridge (2002) |  {{MR|1873379}} | {{ZBL|0992.60001}}</TD></TR>
 
<TR><TD valign="top">[3]</TD> <TD valign="top">Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995)  |  {{MR|1321597}} | {{ZBL|0819.04002}}</TD></TR>
 
<TR><TD valign="top">[3]</TD> <TD valign="top">Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995)  |  {{MR|1321597}} | {{ZBL|0819.04002}}</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989) | {{MR|0982264}} | {{ZBL|0686.60001}}</TD></TR>
+
<TR><TD valign="top">[4]</TD> <TD valign="top">Howard Becker and Alexander S. Kechris, "The descriptive set theory of Polish group actions", Cambridge (1996) | {{MR|1425877}} | {{ZBL|0949.54052}}</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">Paul R. Halmos, "Measure theory", v. Nostrand (1950) | {{MR|0033869}} | {{ZBL|0040.16802}}</TD></TR>
+
<TR><TD valign="top">[5]</TD> <TD valign="top">Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989) | {{MR|0982264}} | {{ZBL|0686.60001}}</TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top">Walter Rudin, "Principles of mathematical analysis", McGraw-Hill (1953) | {{MR|0055409}} | {{ZBL|0052.05301}}</TD></TR>
+
<TR><TD valign="top">[6]</TD> <TD valign="top">Paul R. Halmos, "Measure theory", v. Nostrand (1950) | {{MR|0033869}} | {{ZBL|0040.16802}}</TD></TR>
 +
<TR><TD valign="top">[7]</TD> <TD valign="top">Walter Rudin, "Principles of mathematical analysis", McGraw-Hill (1953) | {{MR|0055409}} | {{ZBL|0052.05301}}</TD></TR>
 
</table>
 
</table>
  
 
[[Category:Classical measure theory]]
 
[[Category:Classical measure theory]]

Revision as of 14:23, 25 December 2011

Also: Borel space

2020 Mathematics Subject Classification: Primary: 28A05 Secondary: 03E1554H05 [MSN][ZBL]

$ \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\Om}{\Omega} \newcommand{\A}{\mathcal A} \newcommand{\B}{\mathcal B} \newcommand{\P}{\mathbf P} $ A measurable space is a set with a distinguished σ-algebra of subsets (called measurable). More formally, it is a pair $(X,\A)$ consisting of a set $X$ and a σ-algebra $\A$ of subsets of $X$.

Examples: $\R^n$ with the Borel σ-algebra; $\R^n$ with the Lebesgue σ-algebra.

Let $(X,\A)$ and $(Y,\B)$ be measurable spaces.

  • A map $f:X\to Y$ is called measurable if $f^{-1}(B) \in \A$ for every $B\in\B$.
  • These two measurable spaces are called isomorphic if there exists a bijection $f:X\to Y$ such that $f$ and $f^{-1}$ are measurable (such $f$ is called an isomorphism).

Let $X$ be a set, $(Y,\B)$ a measurable space, and $(f_i)_{i\in I}$ a family of maps $f_i:X\to Y$. The σ-algebra generated by these maps is defined as the smallest σ-algebra $\A$ on $X$ such that all $f_i$ are measurable from $(X,\A)$ to $(Y,\B)$. More generally, one may take measurable spaces $(Y_i,\B_i)$ and maps $f_i:X\to Y_i$. On the other hand, if $Y$ is $\R$ (or $\C$, $\R^n$ etc.) then $\B$ is by default the Borel σ-algebra.

Given a family of measurable spaces $(X_i,\A_i)$ for $i\in I$, their product is defined as the measurable space $(X,\A)$ where $X=\prod_i X_i$ is the direct product of sets, and $\A$ is generated by the projection maps $p_i:X\to X_i$ [3, Sect. 10.B].

Example and warning. Denoting the Borel σ-algebra on $\R^n$ by $ \B_n $ and the Lebesgue σ-algebra on $\R^n$ by $ \A_n $ we have $ (\R^m,\B_m) \times (\R^n,\B_n) = (\R^{m+n},\B_{m+n}) $ but $ (\R^m,\A_m) \times (\R^n,\A_n) \ne (\R^{m+n},\A_{m+n}) $ [1, Exercise 1.7.19]. Denoting $ (\R^m,\A_m) \times (\R^n,\A_n) = (\R^{m+n},\A_{m,n}) $ we have for every $ A \in \A_{m,n} $ \begin{align*} &\{y\in\R^n:(x,y)\in A\} \in \B_n \quad \text{for almost all } x\in\R^m,\\ &\{x\in\R^m:(x,y)\in A\} \in \B_m \quad \text{for almost all } y\in\R^n. \end{align*} In particular, a set of the form $ \{(x,y)\in\R^2:x-y\in B\} $ belongs to $ \A_{1,1}$ if and only if $B\in\B_1$.

Relations to topological spaces

Every topology generates a σ-algebra, called Borel σ-algebra. That is, the Borel σ-algebra on a topological space is, by definition, generated by the open sets.

Example. The following three σ-algebras on a separable Hilbert space $H$ are equal:

  • the σ-algebra generated by the linear functionals $ x \mapsto \langle x,y \rangle $ for $y\in H$;
  • the Borel σ-algebra corresponding to the norm topology on $H$;
  • the Borel σ-algebra corresponding to the weak topology on $H$.

That is instructive: topological spaces are not a prerequisite to measurable spaces.

A Borel measurable map is generally not continuous, and a Borel isomorphism is generally not a homeomorphism. However, every Borel measurable homomorphism between Polish groups is continuous. Accordingly, the topology of a Polish group is uniquely determined by its Borel σ-algebra (see [3, Sect. 9.C], [4, Sect. 1.2]).

Example. Consider again a separable Hilbert space $H$ and its Borel σ-algebra $\B$. A set $U\subset H$ contains a neighborhood of the origin if and only if there exist $A_1,A_2,\dots\in\B$ such that $A_1\cup A_2\cup\dots=H$ and $A_n-A_n\subset U$ for all $n$ (that is, $x-y\in U$ for all $x,y\in A_n$).

The Borel σ-algebra is not the only bridge between topological and measurable spaces. All sets having the Baire property (sometimes called Baire sets, which may be confusing) are a σ-algebra (generated by open sets together with meager sets) greater than Borel [3, Sect.8.F]. On the other hand, all compact $G_\delta$ subsets of a compact Hausdorff topological space generate a σ-algebra (smaller than Borel) of sets called Baire sets in [5, Sect.7.1], [6, Sect.51]. For more general (in particular, uncountable discrete) topological spaces the definitions of [5] and [6] disagree. Note also the σ-algebra of universally measurable sets.

Older terminology

Weaker assumptions on $\A$ were usual in the past. For example, according to [6], $\A$ need not contain the whole $X$, it is a σ-ring, not necessarily a σ-algebra. According to [7], a measurable space is not a pair $(X,\A)$ but a measure space $(X,\A,\mu)$ such that $X\in\A$ (and again, $\A$ is generally a σ-ring).

References

[1] Terence Tao, "An introduction to measure theory", AMS (2011) | MR2827917 | Zbl 05952932
[2] David Pollard, "A user's guide to measure theoretic probability", Cambridge (2002) | MR1873379 | Zbl 0992.60001
[3] Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995) | MR1321597 | Zbl 0819.04002
[4] Howard Becker and Alexander S. Kechris, "The descriptive set theory of Polish group actions", Cambridge (1996) | MR1425877 | Zbl 0949.54052
[5] Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989) | MR0982264 | Zbl 0686.60001
[6] Paul R. Halmos, "Measure theory", v. Nostrand (1950) | MR0033869 | Zbl 0040.16802
[7] Walter Rudin, "Principles of mathematical analysis", McGraw-Hill (1953) | MR0055409 | Zbl 0052.05301
How to Cite This Entry:
Measurable space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measurable_space&oldid=19942
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article