Simple Function

Definition

$f(\omega )=\sum _{i=1}^n{\alpha }_i{1}_{A_i}(\omega )$ where $A_i\in \mathcal{F}$ are disjoint measurable sets, $a_i$ are the sequence of real or complex numbers, and $1_A$ is a Indicator Function

Finite linear combination of indicator functions of measurable sets

Facts

Simple function $f$ is $\mathcal{F}-\mathcal{R}$ measurable map

A Measurable Function $f$ with a finite Image is a Simple Function