Simple function

A simple function on a measurable space $(X,\Sigma)$ is a measurable function $s:X\to \mathbb R$ (or into $[0,\infty]$) that takes only finitely many distinct values; equivalently, $s$ can be written as $s=\sum_{k=1}^n a_k\,\mathbf 1_{A_k}$ for some real numbers $a_k$ and some measurable sets $A_k\in\Sigma$.

Simple functions are the standard starting point for defining integration (they are finite linear combinations of indicator functions). More complicated measurable functions are often approximated by increasing sequences of simple functions.