I was trying to calculate the Lebesgue integral of a function using $\int_S f\,d\mu = \sum_{k=1}^n a_k \,\mu(D_k\cap S)$, but I need to show that the function is indeed simple function first.
The definition of simple function: $f(x) = \sum_{k=1}^n a_k 1_{D_k}(x)$
I tried writing out $D_k$ and represent $a_k$ using the function, but it turned out that $a_k$ can be hard to generalize with an equation. Are there any ways to show that $a_k$ does exist?
Or are there any ways to show that a function is simple function without specifying $D_k$ and $a_k$?
Reposting my comment as an answer since it appeared satisfactory.
This function is by definition simple: provided $D_k \cap S$ is measurable, $f$ is a linear combination of characteristic functions of measurable sets, and hence is a simple function. Unless you're using a different definition, the choice of $D_k$, $D_k \cap S$, or any other measurable set should be immaterial.