# The approximation of a bounded function by simple functions

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 $f$?

• I wonder if the second sentence could be broken into 2 sentences? Or, is there a missing word, perhaps "if" between "but" and "we"? – Jonas Meyer Jan 1 '13 at 7:14

The limit of a sequence of measurable functions is measurable. If $f$ is not measurable, then there can be no sequence of measurable functions converging pointwise to it.
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 $\varphi_n\rightarrow f$?