Simple function.

[Lekkoksung]

1). Prove that if $\phi$ is a simple function on a measurable set $E$, then $\phi$ is measurable function on $E$.

[Lekkoksung]

ช่วยเช็คให้หน่อยครับ

Let $c \in \mathbb{R}$
Assume that $\phi$ is a simple function on measurable set $E$. Then
$$\phi = \sum_{i=1}^{n}c_{i}\chi_{E_{i}}$$
where $c_{i}$ is a constant, $\cup_{i=1}^{n}E_{i}=E$ and $E_{i} \cap E_{j} = \emptyset$ for all $i \neq j$.
Since $\{ x \in E : \phi(x)= \sum_{i=1}^{n}c_{i}\chi_{E_{i}}(x) > c \}= \{ x \in E_{i} : \phi(x)= c_{i}>c \}=E_{i}$, $\phi$ is a measurable function on E.