Given a measure space $\Omega$.
Consider plain functions $f$.
Define the plain integral as usual: $$f\geq0:\quad\int f\mathrm{d}\mu:=\sup_{s\leq f}\int s\mathrm{d}\mu$$
Do vanishing integrals imply measurability: $$\int|f-s_n|\mathrm{d}\mu\to0\implies s_{n_k}\to f\implies f\in\mathcal{M}$$ (Be aware that the usual proof heavily depends upon measurability!)
Have a guess! ;)
