What is simple function?

A simple function is a measurable function with finitely many values: $s = \sum_{i=1}^n c_i\,\mathbf 1_{A_i}$, $A_i \in \mathcal A$ disjoint, $c_i \geq 0$ (for the nonnegative theory). Its integral is

(convention $0\cdot\infty = 0$); the value does not depend on the representation (refine two partitions).