Editing Measurable function (section)

== Formal definition ==

Let <math>(X,\Sigma)</math> and <math>(Y,\Tau)</math> be measurable spaces, meaning that <math>X</math> and <math>Y</Math> are sets equipped with respective [[σ-algebra|<math>\sigma</math>-algebras]] <math>\Sigma</math> and <math>\Tau.</math>  A function <math>f:X\to Y</math> is said to be measurable if for every <math>E\in \Tau</math> the pre-image of <math>E</math> under <math>f</math> is in <math>\Sigma</math>; that is, for all <math>E \in \Tau </math>
<math display="block">f^{-1}(E) := \{ x\in X \mid f(x) \in E \} \in \Sigma.</math>

That is, <math>\sigma (f)\subseteq\Sigma,</math> where <math>\sigma (f)</math> is the [[Σ-algebra#σ-algebra_generated_by_a_function|σ-algebra generated by f]]. If <math>f:X\to Y</math> is a measurable function, one writes
<math display="block">f \colon (X, \Sigma)  \rightarrow (Y, \Tau).</math>
to emphasize the dependency on the <math>\sigma</math>-algebras <math>\Sigma</math> and <math>\Tau.</math>