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Originally, a measurable function was understood to be a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632001.png" /> of a real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632002.png" /> with the property that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632003.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632004.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632005.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632006.png" /> is a (Lebesgue-) [[Measurable set|measurable set]]. A measurable function on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632007.png" /> can be made continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m0632008.png" /> by changing its values on a set of arbitrarily small measure; this is the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320010.png" />-property of measurable functions (N.N. Luzin, 1913, cf. also [[Luzin-C-property|Luzin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320011.png" />-property]]).
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A measurable function on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320012.png" /> is defined relative to a chosen system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320013.png" /> of measurable sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320015.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320016.png" />-ring, then a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320017.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320018.png" /> is said to be a measurable function if
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{{TEX|done}}
  
<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/m063200/m06320019.png" /></td> </tr></table>
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{{MSC|28A20}}
  
for every real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320020.png" />, where
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[[Category:Classical measure theory]]
  
<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/m063200/m06320021.png" /></td> </tr></table>
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Originally, a measurable function was understood to be a function  $  f ( x) $
 +
of a real variable  $  x $
 +
with the property that for every  $  a $
 +
the set  $  E _ {a} $
 +
of points  $  x $
 +
at which  $  f ( x) < a $
 +
is a (Lebesgue-) [[Measurable set|measurable set]]. A measurable function on an interval  $  [ x _ {1} , x _ {2} ] $
 +
can be made continuous on  $  [ x _ {1} , x _ {2} ] $
 +
by changing its values on a set of arbitrarily small measure; this is the so-called  $  C $-
 +
property of measurable functions (N.N. Luzin, 1913, cf. also [[Luzin-C-property|Luzin  $  C $-
 +
property]]).
  
<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/m063200/m06320022.png" /></td> </tr></table>
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A measurable function on a space  $  X $
 +
is defined relative to a chosen system  $  A $
 +
of measurable sets in  $  X $.
 +
If  $  A $
 +
is a  $  \sigma $-
 +
ring, then a real-valued function  $  f $
 +
on  $  X $
 +
is said to be a measurable function if
  
This definition is equivalent to the following: A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320023.png" /> is measurable if
+
$$
 +
R _ {f} \cap E _ {a}  \in  A
 +
$$
  
<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/m063200/m06320024.png" /></td> </tr></table>
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for every real number  $  a $,
 +
where
  
for every [[Borel set|Borel set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320025.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320026.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320027.png" />-algebra, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320028.png" /> is measurable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320029.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320030.png" />) is measurable. The class of measurable functions is closed under the arithmetical and lattice operations; that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320032.png" /> are measurable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320037.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320038.png" /> real) are measurable; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063200/m06320040.png" /> are also measurable. A complex-valued function is measurable if its real and imaginary parts are measurable. A generalization of the concept of a measurable function is that of a [[Measurable mapping|measurable mapping]] from one [[Measurable space|measurable space]] to another.
+
$$
 +
E _ {a}  =  \{ {x \in X } : {f ( x) < a } \}
 +
,
 +
$$
 +
 
 +
$$
 +
R _ {f}  =  \{ x \in X:  f ( x) \neq 0 \} .
 +
$$
 +
 
 +
This definition is equivalent to the following: A real-valued function  $  f $
 +
is measurable if
 +
 
 +
$$
 +
R _ {f} \cap \{ {x \in X } : {f ( x) \in B } \}
 +
\in  A
 +
$$
 +
 
 +
for every [[Borel set|Borel set]] $  B $.  
 +
When $  A $
 +
is a $  \sigma $-
 +
algebra, a function $  f $
 +
is measurable if $  E _ {a} $(
 +
or $  \{ {x \in X } : {f ( x) \in B } \} $)  
 +
is measurable. The class of measurable functions is closed under the arithmetical and lattice operations; that is, if $  f _ {n} $,
 +
$  n = 1, 2 \dots $
 +
are measurable, then $  f _ {1} + f _ {2} $,
 +
$  f _ {1} f _ {2} $,
 +
$  \max ( f _ {1} , f _ {2} ) $,  
 +
$  \min ( f _ {1} , f _ {2} ) $
 +
and $  af $(
 +
$  a $
 +
real) are measurable; $  \overline{\lim\limits}\;  f _ {n} $
 +
and $  fnnme \underline{lim}  f _ {n} $
 +
are also measurable. A complex-valued function is measurable if its real and imaginary parts are measurable. A generalization of the concept of a measurable function is that of a [[Measurable mapping|measurable mapping]] from one [[Measurable space|measurable space]] to another.
  
 
====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"N. Dunford,   J.T. Schwartz,   "Linear operators. General theory" , '''1''' , Interscience (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,   "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock (1957–1961) (Translated from Russian)</TD></TR></table>
+
{|
 +
|valign="top"|{{Ref|H}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
 +
|-
 +
|valign="top"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958) {{MR|0117523}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|KF}}|| A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock (1957–1961) (Translated from Russian) {{MR|1025126}} {{MR|0708717}} {{MR|0630899}} {{MR|0435771}} {{MR|0377444}} {{MR|0234241}} {{MR|0215962}} {{MR|0118796}} {{MR|1530727}} {{MR|0118795}} {{MR|0085462}} {{MR|0070045}} {{ZBL|0932.46001}} {{ZBL|0672.46001}} {{ZBL|0501.46001}} {{ZBL|0501.46002}} {{ZBL|0235.46001}} {{ZBL|0103.08801}}
 +
|}

Latest revision as of 08:00, 6 June 2020


2020 Mathematics Subject Classification: Primary: 28A20 [MSN][ZBL]

Originally, a measurable function was understood to be a function $ f ( x) $ of a real variable $ x $ with the property that for every $ a $ the set $ E _ {a} $ of points $ x $ at which $ f ( x) < a $ is a (Lebesgue-) measurable set. A measurable function on an interval $ [ x _ {1} , x _ {2} ] $ can be made continuous on $ [ x _ {1} , x _ {2} ] $ by changing its values on a set of arbitrarily small measure; this is the so-called $ C $- property of measurable functions (N.N. Luzin, 1913, cf. also Luzin $ C $- property).

A measurable function on a space $ X $ is defined relative to a chosen system $ A $ of measurable sets in $ X $. If $ A $ is a $ \sigma $- ring, then a real-valued function $ f $ on $ X $ is said to be a measurable function if

$$ R _ {f} \cap E _ {a} \in A $$

for every real number $ a $, where

$$ E _ {a} = \{ {x \in X } : {f ( x) < a } \} , $$

$$ R _ {f} = \{ x \in X: f ( x) \neq 0 \} . $$

This definition is equivalent to the following: A real-valued function $ f $ is measurable if

$$ R _ {f} \cap \{ {x \in X } : {f ( x) \in B } \} \in A $$

for every Borel set $ B $. When $ A $ is a $ \sigma $- algebra, a function $ f $ is measurable if $ E _ {a} $( or $ \{ {x \in X } : {f ( x) \in B } \} $) is measurable. The class of measurable functions is closed under the arithmetical and lattice operations; that is, if $ f _ {n} $, $ n = 1, 2 \dots $ are measurable, then $ f _ {1} + f _ {2} $, $ f _ {1} f _ {2} $, $ \max ( f _ {1} , f _ {2} ) $, $ \min ( f _ {1} , f _ {2} ) $ and $ af $( $ a $ real) are measurable; $ \overline{\lim\limits}\; f _ {n} $ and $ fnnme \underline{lim} f _ {n} $ are also measurable. A complex-valued function is measurable if its real and imaginary parts are measurable. A generalization of the concept of a measurable function is that of a measurable mapping from one measurable space to another.

References

[H] P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[KF] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) MR1025126 MR0708717 MR0630899 MR0435771 MR0377444 MR0234241 MR0215962 MR0118796 MR1530727 MR0118795 MR0085462 MR0070045 Zbl 0932.46001 Zbl 0672.46001 Zbl 0501.46001 Zbl 0501.46002 Zbl 0235.46001 Zbl 0103.08801
How to Cite This Entry:
Measurable function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measurable_function&oldid=11460
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article