If $f$ is a bounded function (not necessarily measurable), is it true that we can find a sequence of simple functions $\{\varphi_n\}$ such that $\varphi_n\rightarrow f$?
I wonder this because in the definition of Lebesgue integral, we define $\int f$ to be the limit of $\int \varphi_n$ when $\varphi_n\rightarrow f$, and we only assumed $f$ to be bounded and supported on a set of finite measure, but we didn't assume that it was measurable, then can we always find {$\varphi_n$} to define the Lebesgue integral of $f$?