Let $(X,\mathcal{F},\mu)$ be a \reftext{def:measure-measure-space-2026a}{measure space}. For $A\subseteq X$, the \textbf{indicator function} $\mathbf{1}_A:X\to\mathbb{R}$ is defined by $\mathbf{1}_A(x)=1$ for $x\in A$ and $\mathbf{1}_A(x)=0$ otherwise.

A \textbf{simple function} on $(X,\mathcal{F})$ is a \reftext{def:measurable-function-2026a}{measurable} function $s:X\to\mathbb{R}$ that takes only finitely many values. Writing $c_1,\dots,c_r$ for the distinct values of $s$ and $A_i=s^{-1}(\{c_i\})$, the sets $A_1,\dots,A_r$ are measurable (each $\{c_i\}$ is a \reftext{def:borel-sigma-algebra-real-line-2026a}{Borel set}), pairwise disjoint, cover $X$, and

this is the \textbf{standard representation} of $s$.

If $s$ is a nonnegative simple function with standard representation as above, its \textbf{integral} with respect to $\mu$ is

with the conventions of \ref{def:measure-measure-space-2026a}, in particular $0\cdot\infty=0$.