Here is the theorem I want to prove:
I guess long ago when I studied analysis this theorem was given in some textbook with its proof. I think it was Kolomogorov book, or Royden Real analysis fourth edition. I am not sure which one though.
Could someone tell me a source where I can find this theorem with its detailed proof? Or give me the proof.
Let $(X,\mathcal{M}, \mu)$ be a measure space with $\mu(X) < \infty,$ and let $(X,\overline{\mathcal{M}}, \overline{\mu})$ be its completion. Suppose $f: X \to \mathbb R$ is bounded. Then $f$ is $\overline{\mathcal{M}}$-measurable (and hence in $L^1(\overline{\mu})$) iff there exist sequences $\{\phi_n\}$ and $\{\psi_n\}$ of $\mathcal{M}$-measurable simple functions such that $\phi_n \leq f \leq \psi_n$ and $\int(\psi_n - \phi_n) d\mu < n^{-1}.$ In this case, $\lim \int \phi_n d\mu = \lim \int \psi_n d\mu = \int f d\overline{\mu}.$