Let $(\Omega, \mathscr F, \nu)$ be a measure space and let $f$ and $g$ be non-negative Borel functions. Let $\psi$ be a simple function such that $0 \leq \psi \leq f + g$.
I want to show that it is possible to find simple functions $\phi$ and $\phi'$ with $0 \leq \phi \leq f$ and $0 \leq \phi' \leq g$ such that $$ \int (\psi - \phi - \phi') d\nu < \varepsilon $$ where $\varepsilon > 0$ is arbitrary.
We can represent each simple function as a finite sum $\psi = \sum_{k=1}^N c_k \mathbf 1_{C_k}$, $\phi = \sum_{i=1}^n a_i \mathbf 1_{A_i}$, and $\phi' = \sum_{j=1}^m b_j \mathbf 1_{B_j}$ where each of $\{A_i\}$, $\{B_j\}$, and $\{C_k\}$ partition $\Omega$. This means that $$ \psi - \phi - \phi' = \sum_{ijk} (c_k - a_i - b_j) \mathbf 1_{A_i \cap B_j \cap C_k} $$ so I need to make $$ \int (\psi - \phi - \phi')d\nu = \sum_{ijk} (c_k - a_i - b_j) \nu(A_i \cap B_j \cap C_k) $$ arbitrarily small.
What I'm struggling with is how to do this while also keeping the $\phi \leq f$ and the $\phi' \leq g$ constraints. All of the other proofs involving approximations with simple functions that I've seen don't have to deal with this issue.
I'd be happy to see a proof that requires $\int fd\nu$, $\int g d\nu < \infty$ although I'm hoping this result holds even if that is not true.