approximation of measurable functions 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.
