Simple Function

A simple function on $X$ is a function $s:X\to \mathbb{R}$ of the form $s={\Sigma }_{i=1}^n{a}_i\times X_{a_i}$ for pairwise disjoint $A_i\subset X$ and distinct real numbers $a_i$

$X_A(x)=\{\begin{array}{ll}1& \text{If}x\in A\\ 0& \text{If}x\notin A\end{array}$

Note

If $X$ has $M$ then: $s$ is measurable $⟺$ all $A_i$ are measurable

$A_i=s^{-1}(\{a_i\})=s^{-1}(\mathbb{R}\setminus \{a_i\}{)}^{\complement }$

If we have a measure $\mu$ on $X$, define ${\int }_{A}sd\mu \equiv {\Sigma }_{i=1}^n{a}_i\times \mu (A\cap A_i)$ for $A\in M$ (they are all $\in [0,\infty ])$.