# Measurable function

2010 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.

How to Cite This Entry:
Measurable function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Measurable_function&oldid=47814
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article