Simple function

Remarks

Equivalently, $s = \sum_{k=1}^{n} a_k \mathbf{1}_{A_k}$ for some $n \ge 0$, real numbers $a_k$ and sets $A_k \in \Sigma$, where $\mathbf{1}_{A_k}$ is the indicator function of $A_k$. The sets $A_k$ can be chosen pairwise disjoint, with distinct values $a_k$. For a measure $\mu$ and $a_k \ge 0$, the integral is $\int s \, d\mu = \sum_k a_k \mu(A_k)$, with $0 \cdot \infty = 0$. Example: on $\R$ with the Borel sets, $s = 3 \cdot \mathbf{1}_{[0, 1]} - \mathbf{1}_{\Q}$ is simple; its values are $-1$, $0$, $2$ and $3$. Non-example: $x \mapsto x$ on $[0, 1]$ is measurable but has infinitely many values. Every measurable $f \colon X \to [0, \infty]$ is the pointwise limit of a non-decreasing sequence of nonnegative simple functions. Some authors drop the measurability condition, or allow complex values.