Hi we know we can approximate measurable function by simple function however can we increase the conditions such that we can approximate by at most countable functions that is there exists sequence {$s_n$} of non negative F/B* measurable functions such that each assumes at most countable many values, all values finite and $s_n \rightarrow$ f """uniformly""" ? I tried to change the proof for approximation of simple functions given below however I am having troubles I don't see how I can get the result for those extended simple functions.
Here F/B* are defined as follows F is a sigma field on $\Omega$ and define F/B* as follows: Let A $\in$ F be nonempty, and let f : A $\rightarrow$ $R^{*}$ denote a function.
We will say that f is F/B*-measurable iff $f^{-1}(B)$ $\in$ F and both $f^{-1}$({$\infty$}) and $f^{-1}$({-$\infty$}) are in F.
For $n\in \Bbb{N}$ and $k\in \Bbb{Z}$, let $M_n^k = f^{-1}((k/n, (k+1)/n])$. Define
$$ s_n := \sum_{k\in \Bbb{Z}} \frac{k+1}{n} \chi_{M_n^k}, $$ where $\chi_M$ is s the characteristic/indicator function of the set $M$.
I leave it to you to verify $\Vert f -s_n\Vert_\sup \leq 1/n$ and hence $s_n \to f$ uniformly.
EDIT: If $f$ only assumed nonnegative values, then so does $s_n$. If $f$ is also allowed to assume $\infty$, then $s_n \to f$ uniformly is not possible if $s_n$ should only assume finite values.