Definition:Standard Representation of Simple Function

Definition
Let $\left({X, \Sigma}\right)$ be a measurable space.

Let $f: X \to \R$ be a simple function.

A standard representation of $f$ consists of:


 * a finite sequence $a_1, \ldots, a_n$ of real numbers
 * a finite sequence $E_1, \ldots, E_n$ of pairwise disjoint, $\Sigma$-measurable sets

subject to:


 * $f = \displaystyle \sum_{j=0}^n a_j \chi_{E_j}$

where $a_0 := 0$, $E_0 := X \setminus \left({\displaystyle \bigcup_{j = 1}^n E_j}\right)$ and $\chi$ denotes characteristic function.