Definition:Simple Function

General Definition
Let $\left({X, \mathcal A, \mu}\right)$ be a measure space.

A mapping $\phi: X \to \R$ is said to be simple if it is a finite linear combination of characteristic functions:
 * $\displaystyle \phi = \sum_{k=1}^n a_k \chi_{S_k}$

where $a_1, a_2, \ldots, a_n$ are real numbers and each of the sets $S_k$ is $\mathcal A$-measurable.

Definition for Real Functions
A real function $\phi: \R \to \R$ is said to be simple if it is a finite linear combination of characteristic functions:
 * $\displaystyle \phi = \sum_{k=1}^n a_k \chi_{S_k}$

where $a_1, a_2, \ldots, a_n$ are real numbers and each of the sets $S_k$ is (Lebesgue) measurable.