Question about simple functions

Question

Based on Williams' Probability w/ Martingales:

Let $(S, \Sigma, \mu)$ be a measure space. A $\Sigma$-meas function f is called simple if f may be written as a finite sum $f = \sum_{k=1}^{m} a_k I_{A_k}$ where $a_k \ge 0$ and $A_k \in \Sigma$

Need it be that $\bigcup_{k} A_k = S$ or $\bigcup_{k} A_k =$ Domain of f?

Answer 1

We can assume without loss of generality that $\bigcup_{k} A_k = S$.

Indeed: If the $A_k$ do not satisfy this condition, we define $\alpha_0 := 0$ and

$$A_0 := S \backslash \bigcup_{k=1}^m A_k.$$

Then,

$$f = \sum_{k=1}^m \alpha_k I_{A_k} + 0 \cdot 1_{A_0} = \sum_{k=0}^m \alpha_k I_{A_k}.$$