Simple Function and Its Integral

Let $(X,\mathcal{F},\mu)$ be a measure space. For $A\subseteq X$, the 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 simple function on $(X,\mathcal{F})$ is a 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 Borel set), pairwise disjoint, cover $X$, and

this is the standard representation of $s$.

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

with the conventions of Measure, Measure Space, and Probability Measure, in particular $0\cdot\infty=0$.