Editing Measurable function (section)

== Notable classes of measurable functions ==

* Random variables are by definition measurable functions defined on probability spaces.
* If <math>(X, \Sigma)</math> and <math>(Y, T)</math> are [[Borel set#Standard Borel spaces and Kuratowski theorems|Borel space]]s, a measurable function <math>f:(X, \Sigma) \to (Y, T)</math>  is also called a '''Borel function'''. Continuous functions are Borel functions but not all Borel functions are continuous. However, a measurable function is nearly a continuous function; see [[Luzin's theorem]].  If a Borel function happens to be a section of a map <math>Y\xrightarrow{~\pi~}X,</math> it is called a '''Borel section'''.
* A [[Lebesgue measurable]] function is a measurable function <math>f : (\R, \mathcal{L}) \to (\Complex, \mathcal{B}_\Complex),</math> where <math>\mathcal{L}</math> is the <math>\sigma</math>-algebra of Lebesgue measurable sets, and <math>\mathcal{B}_\Complex</math> is the [[Borel algebra]] on the [[complex number]]s <math>\Complex.</math>  Lebesgue measurable functions are of interest in [[mathematical analysis]] because they can be integrated. In the case <math>f : X \to \R,</math> <math>f</math> is Lebesgue measurable if and only if <math>\{f > \alpha\} = \{ x\in X : f(x) > \alpha\}</math> is measurable for all <math>\alpha\in\R.</math> This is also equivalent to any of <math>\{f \geq \alpha\},\{f<\alpha\},\{f\le\alpha\}</math> being measurable for all <math>\alpha,</math> or the preimage of any open set being measurable. Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functions of bounded variation are all Lebesgue measurable.<ref name="carothers">{{cite book |last=Carothers|first=N. L.|title=Real Analysis|url=https://archive.org/details/realanalysis0000caro| url-access=registration | year=2000| publisher=Cambridge University Press| isbn=0-521-49756-6}}</ref> A function <math>f:X\to\Complex</math> is measurable if and only if the real and imaginary parts are measurable.