Simple function (measure theory)/Definition

Definition

A simple function $f:X\rightarrow\mathbb{R}$ on a measurable space $(X,\mathcal{A})$ is a:

• function of the form $\sum^N_{i=1}x_i\mathbf{1}_{A_i}(x)$ for
• finitely many sets, $A_1,\ldots,A_N\in\mathcal{A}$ and
• finitely many $x_1,\ldots,x_n\in\mathbb{R}$