Uniform Approximation of Continuous Functions by Nontrivial Simple Functions

Abstract

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.