Definition (Simple function)

A function $f:X\to \mathbb{R}$ is called simple if:

• $f$ has a finite range (i.e. $|X|<\infty$) and;
• $f:(X,\mathcal{F})\to(\mathbb{R},\mathcal{B}(\mathbb{R}))$ is measurable

i.e. for $A_{i}\in\mathcal{F}$ we can express $f$ as: $$f(X)=\sum_{i=1}^{n}a_{i}\mathbb{1}_{A_{i}}$$where $a_{i}\ge0$ and $f(X) = \{ a_{1},\dots,a_{m} \}$. The set of simple functions is denoted as $S^{+}$.