  We prove that every nonnegative continuous real-valued function on a given
compact metric space is the uniform limit of some increasing sequence of
nonnegative simple functions being linear combinations of indicators of open
sets; here the nontriviality is relative to the standard choice(s) of
approximating simple functions for measurable functions, where one loses
control over the indicated measurable sets. Thus the standard uniform
approximation of bounded nonnegative measurable real-valued functions by
increasing nonnegative simple functions may be improved for nonnegative
continuous real-valued functions on compact metric spaces. There are also some
interesting consequences regarding semi-continuous functions and smooth
functions.
