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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m0632101.png" /> of a [[Measurable space|measurable space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m0632102.png" /> to a measurable space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m0632103.png" /> such that
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A mapping $f$ of a [[Measurable space|measurable space]] $(X_1, \mathcal A_1)$ to a measurable space $(X_2, \mathcal A_2)$ such that
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m0632104.png" /></td> </tr></table>
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f^{-1}(A) = \{x : f(x)\in A\} \in \mathcal A_1 \quad \text{ for each } A\in\mathcal A_2.
 
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$$
In the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m0632105.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m0632106.png" />-algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m0632107.png" /> is the real line with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m0632108.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m0632109.png" /> of Borel sets (cf. [[Borel set|Borel set]]), the concept of a measurable mapping reduces to that of a [[Measurable function|measurable function]] (however, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321010.png" /> is only a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321011.png" />-ring, the definition of a measurable function is usually modified in accordance with the requirements of integration theory). The superposition of measurable mappings is measurable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321013.png" /> are rings and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321014.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321015.png" /> in some class of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321016.png" /> such that the ring generated by it is the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321017.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321018.png" /> is measurable. The analogous assertions hold in the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321019.png" />-rings, algebras and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321020.png" />-algebras. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321022.png" /> are topological spaces with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321023.png" />-algebras of Borel sets, then every continuous mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321024.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321025.png" /> is measurable. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321026.png" /> be a topological space, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321027.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321028.png" />-algebra of Borel sets and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321029.png" /> be a finite non-negative regular measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321030.png" /> (regularity means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321031.png" />). Suppose further that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321032.png" /> is a separable metric space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321033.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321034.png" />-algebra of Borel sets, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321035.png" /> be a measurable mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321036.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321037.png" />. Then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321038.png" /> there is a closed subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321039.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321041.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063210/m06321042.png" /> (Luzin's theorem).
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In the case where $\mathcal A_1$ is a $\sigma$-algebra and $(X_2, \mathcal A_2)$ is the real line with the $\sigma$-algebra $\mathcal A_2$ of Borel sets (cf. [[Borel set|Borel set]]), the concept of a measurable mapping reduces to that of a [[Measurable function|measurable function]] (however, when $\mathcal A_1$ is only a $\sigma$-ring, the definition of a measurable function is usually modified in accordance with the requirements of integration theory). The superposition of measurable mappings is measurable. If $\mathcal A_1$ and $\mathcal A_2$ are rings and $f^{-1}(B)\in\mathcal A_1$ for each $B$ in some class of sets $B\in\mathcal A_2$ such that the ring generated by it is the whole of $\mathcal A_2$, then $f$ is measurable. The analogous assertions hold in the case of $\sigma$-rings, algebras and $\sigma$-algebras. If $(X_1, \mathcal A_1)$ and $(X_2, \mathcal A_2)$ are topological spaces with the $\sigma$-algebras of Borel sets, then every continuous mapping from $X_1$ to $X_2$ is measurable. Let $X$ be a topological space, let $\mathcal A$ be the $\sigma$-algebra of Borel sets and let $\mu$ be a finite non-negative regular measure on $\mathcal A$ (regularity means that $\mu(A) = \sup \{\mu(F) : F\subset A, F \text{ closed}\}$). Suppose further that $Y$ is a separable metric space, $\mathcal B$ is the $\sigma$-algebra of Borel sets, and let $f$ be a measurable mapping from $(X, \mathcal A)$ to $(Y, \mathcal B)$. Then for any $\varepsilon>0$ there is a closed subset $F\subset X$ such that $\mu(X\setminus F)<\varepsilon$ and $f$ is continuous on $F$ (Luzin's theorem).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Neveu,  "Mathematical foundations of the calculus of probabilities" , Holden-Day  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Neveu,  "Mathematical foundations of the calculus of probabilities" , Holden-Day  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR></table>

Latest revision as of 06:33, 23 July 2015

A mapping $f$ of a measurable space $(X_1, \mathcal A_1)$ to a measurable space $(X_2, \mathcal A_2)$ such that $$ f^{-1}(A) = \{x : f(x)\in A\} \in \mathcal A_1 \quad \text{ for each } A\in\mathcal A_2. $$ In the case where $\mathcal A_1$ is a $\sigma$-algebra and $(X_2, \mathcal A_2)$ is the real line with the $\sigma$-algebra $\mathcal A_2$ of Borel sets (cf. Borel set), the concept of a measurable mapping reduces to that of a measurable function (however, when $\mathcal A_1$ is only a $\sigma$-ring, the definition of a measurable function is usually modified in accordance with the requirements of integration theory). The superposition of measurable mappings is measurable. If $\mathcal A_1$ and $\mathcal A_2$ are rings and $f^{-1}(B)\in\mathcal A_1$ for each $B$ in some class of sets $B\in\mathcal A_2$ such that the ring generated by it is the whole of $\mathcal A_2$, then $f$ is measurable. The analogous assertions hold in the case of $\sigma$-rings, algebras and $\sigma$-algebras. If $(X_1, \mathcal A_1)$ and $(X_2, \mathcal A_2)$ are topological spaces with the $\sigma$-algebras of Borel sets, then every continuous mapping from $X_1$ to $X_2$ is measurable. Let $X$ be a topological space, let $\mathcal A$ be the $\sigma$-algebra of Borel sets and let $\mu$ be a finite non-negative regular measure on $\mathcal A$ (regularity means that $\mu(A) = \sup \{\mu(F) : F\subset A, F \text{ closed}\}$). Suppose further that $Y$ is a separable metric space, $\mathcal B$ is the $\sigma$-algebra of Borel sets, and let $f$ be a measurable mapping from $(X, \mathcal A)$ to $(Y, \mathcal B)$. Then for any $\varepsilon>0$ there is a closed subset $F\subset X$ such that $\mu(X\setminus F)<\varepsilon$ and $f$ is continuous on $F$ (Luzin's theorem).

References

[1] P.R. Halmos, "Measure theory" , v. Nostrand (1950)
[2] J. Neveu, "Mathematical foundations of the calculus of probabilities" , Holden-Day (1965) (Translated from French)
[3] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)
[4] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
How to Cite This Entry:
Measurable mapping. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Measurable_mapping&oldid=36560
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article