(Solved): Exercise 2. (3p) By \( \chi_{E} \) we denote the characteristic function associated with the set \( ...

Exercise 2. (3p) By \( \chi_{E} \) we denote the characteristic function associated with the set \( E \). Show that the function \( \varphi \) defined below is simple (you may quote a theorem for checking the 'measurable' part, but precisely find \( \operatorname{Im} \varphi \) ). \[ \varphi: \mathbb{R} \rightarrow \mathbb{R}, \quad \varphi(x)=5 \chi_{[0,3]}(x)-2 \chi_{[1,4]}(x) \]