The title says it all. How can we approximate measurable simple functions by step functions almost uniformly in, say, $[0,1]$? Even with the simplest example, $\chi_{A}$, where $A$ is Lebesgue measurable, I cannot do this almost everywhere.
I tried moving to decreasing intersections of open sets, which works out fine, but then moving to finite intersections gives you sets of positive measure where the approximation in norm is not small anymore.
Use Lusin's Theorem to find a continuous function that agrees with your simple function off a set of measure less than $\varepsilon$. Now use compactness of $[0,1]$ to write the continuous function as a uniform limit of step functions.