Uniform Approximation of Continuous Functions by Nontrivial Simple Functions

TL;DR

This paper demonstrates that continuous nonnegative functions on compact metric spaces can be uniformly approximated by increasing sequences of simple functions composed of open set indicators, improving upon standard methods.

Contribution

It introduces a novel uniform approximation approach for continuous functions using nontrivial simple functions, enhancing classical approximation techniques.

Findings

Every nonnegative continuous function can be uniformly approximated by increasing simple functions.

The approximation method improves upon standard approaches for measurable functions.

Implications for semi-continuous and smooth functions are discussed.

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.