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.

Examples:

• A step function on $\mathbb R$ that is constant on finitely many intervals and zero elsewhere is a simple function (with respect to the Borel sigma-algebra).
• If $A,B$ are measurable sets in $(X,\Sigma)$, then $2\,\mathbf 1_A - 3\,\mathbf 1_B$ is a simple function.