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Difference between pages "Measurable space" and "Category:Analysis"

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''Borel space''
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* {{User:Rehmann/sandbox/MSCtop|26}} [[:Category:Real functions]] {See also 54C30}
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* {{User:Rehmann/sandbox/MSCtop|28}} [[:Category:Measure and integration]] {For analysis on manifolds, see 58-XX}
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* {{User:Rehmann/sandbox/MSCtop|30}} [[:Category:Functions of a complex variable]] {For analysis on manifolds, see 58-XX}
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* {{User:Rehmann/sandbox/MSCtop|31}} [[:Category:Potential theory]] {For probabilistic potential theory, see 60J45}
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* {{User:Rehmann/sandbox/MSCtop|32}} [[:Category:Several complex variables and analytic spaces]] {For infinite-dimensional holomorphy, see 46G20, 58B12}
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* {{User:Rehmann/sandbox/MSCtop|33}} [[:Category:Special functions]] (33-XX deals with the properties of functions as functions) {For orthogonal functions, see 42Cxx; for aspects of combinatorics, see 05Axx; for number-theoretic aspects, see 11-XX; for representation theory, see 22Exx}
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* {{User:Rehmann/sandbox/MSCtop|40}} [[:Category:Sequences, series, summability]]
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* {{User:Rehmann/sandbox/MSCtop|41}} [[:Category:Approximations and expansions]] {For all approximation theory in the complex domain, see 30E05 and 30E10; for all trigonometric approximation and interpolation, see 42A10 and 42A15; for numerical approximation, see 65Dxx}
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* {{User:Rehmann/sandbox/MSCtop|42}} [[:Category:Harmonic analysis on Euclidean spaces]]
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* {{User:Rehmann/sandbox/MSCtop|43}} [[:Category:Abstract harmonic analysis]] {For other analysis on topological and Lie groups, see 22Exx}
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* {{User:Rehmann/sandbox/MSCtop|44}} [[:Category:Integral transforms, operational calculus]] {For fractional derivatives and integrals, see 26A33. For Fourier transforms, see 42A38, 42B10. For integral transforms in distribution spaces, see 46F12. For numerical methods, see 65R10}
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* {{User:Rehmann/sandbox/MSCtop|45}} [[:Category:Integral equations]]
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* {{User:Rehmann/sandbox/MSCtop|46}} [[:Category:Functional analysis]] {For manifolds modeled on topological linear spaces, see 57Nxx, 58Bxx}
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* {{User:Rehmann/sandbox/MSCtop|47}} [[:Category:Operator theory]]
  
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{{User:Boris Tsirelson/MSC|28A05|03E15,54H05}}
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$ \newcommand{\R}{\mathbb R}
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[[Category:Mathematics]]
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\newcommand{\C}{\mathbb C}
 
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\newcommand{\Om}{\Omega}
 
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\newcommand{\A}{\mathcal A}
 
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\newcommand{\B}{\mathcal B}
 
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\newcommand{\P}{\mathbf P} $
 
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A set with a distinguished [[Algebra of sets|σ-algebra]] of subsets (called measurable). More formally: a pair $(X,\A)$ consisting of a set $X$ and a σ-algebra $\A$ of subsets of $X$.
 
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''Examples:'' $\R^n$ with the [[Borel set|Borel σ-algebra]]; $\R^n$ with the [[Lebesgue measure|Lebesgue σ-algebra]].
 
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Let $(X,\A)$ and $(Y,\B)$ be measurable spaces.
 
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* A map $f:X\to Y$ is called ''measurable'' if $f^{-1}(B) \in \A$ for every $B\in\B$.
 
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* 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).
 
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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.
 
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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].
 
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''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+m},\A_{m,n}) $ we have for every $ A \in \A_{m,n} $
 
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\begin{align*}
 
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&\{y\in\R^n:(x,y)\in A\} \in \B_n \quad \text{for almost all } x\in\R^m,\\
 
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&\{x\in\R^m:(x,y)\in A\} \in \B_m \quad \text{for almost all } y\in\R^n.
 
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\end{align*}
 
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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 $B\in\B_1$.
 
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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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''Example.'' The following three σ-algebras on a separable [[Hilbert space]] $H$ are equal:
 
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* the σ-algebra generated by the linear functionals $ x \mapsto \langle x,y \rangle $ for $y\in H$;
 
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* the Borel σ-algebra corresponding to the norm topology on $H$;
 
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* the Borel σ-algebra corresponding to the weak topology on $H$.
 
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That is instructive: topological spaces are not a prerequisite to measurable spaces.
 
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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
 
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[[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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====Older terminology====
 
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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).
 
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====References====
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">Terence Tao, "An introduction to measure theory", AMS (2011) {{User:Boris Tsirelson/MR|2827917}}</TD></TR>
 
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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) {{User:Boris Tsirelson/MR|1873379}}</TD></TR>
 
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<TR><TD valign="top">[3]</TD> <TD valign="top">Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995) {{User:Boris Tsirelson/MR|1321597}}</TD></TR>
 
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<TR><TD valign="top">[4]</TD> <TD valign="top">Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989) {{User:Boris Tsirelson/MR|0982264}}</TD></TR>
 
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<TR><TD valign="top">[5]</TD> <TD valign="top">Paul R. Halmos, "Measure theory", v. Nostrand (1950) {{User:Boris Tsirelson/MR|0033869}}</TD></TR>
 
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<TR><TD valign="top">[6]</TD> <TD valign="top">Walter Rudin, "Principles of mathematical analysis", McGraw-Hill (1953) {{User:Boris Tsirelson/MR|0055409}}</TD></TR>
 
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</table>
 
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[[Category:Classical measure theory]]
 

Revision as of 22:52, 7 January 2012

(MSC2010 codes)

How to Cite This Entry:
Measurable space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measurable_space&oldid=19887
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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