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{{MSC|28A20}}

[[Category:Classical measure theory]]

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]]).

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|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====
{|
|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}}
|}