If $X=\sum_{i=1}^4a_i\chi_{A_i}$ is a simple function with the $\chi_{A_i}$'s being the indicator functions. Also the$A_i$'s are disjoint, $a_i$'s are distinct, and $\bigcup_{i=1}^4A_i=\Omega$, the whole space. Need to figure out the exact subsets of the $\sigma$-algebra $u(X)$.
So what I did was to take every possible combination of the disjoint sets $A_i$'s. That is,
$$u(X)=\{\emptyset,\{A_i\},\{A_i,A_j\},\{A_i,A_j,A_k\},\Omega\}$$
Is this correct? What roles (if any) do the $a_i\in\mathbb{R}$ play?