Editing Measurable function (section)

== Non-measurable functions ==

Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to prove the existence of non-measurable functions.  Such proofs rely on the [[axiom of choice]] in an essential way, in the sense that [[Zermelo–Fraenkel set theory]] without the axiom of choice does not prove the existence of such functions.

In any measure space ''<math>(X, \Sigma)</math>'' with a [[non-measurable set]] <math>A \subset X,</math> <math>A \notin \Sigma,</math> one can construct a non-measurable [[indicator function]]: 
<math display="block">\mathbf{1}_A:(X,\Sigma) \to \R,
\quad
\mathbf{1}_A(x) = \begin{cases}
1 & \text{ if } x \in A \\
0 & \text{ otherwise},
\end{cases}</math>
where <math>\R</math> is equipped with the usual [[Borel algebra]]. This is a non-measurable function since the preimage of the measurable set <math>\{1\}</math> is the non-measurable <math>A.</math>  

As another example, any non-constant function <math>f : X \to \R</math> is non-measurable with respect to the trivial <math>\sigma</math>-algebra <math>\Sigma = \{\varnothing, X\},</math> since the preimage of any point in the range is some proper, nonempty subset of <math>X,</math> which is not an element of the trivial <math>\Sigma.</math>