Existence of continuous functions that are one-to-one almost everywhere

TL;DR

This paper proves that in any metric space with a suitable measure, there exists a bounded continuous function that is injective except on a measure-zero set, highlighting the richness of continuous functions.

Contribution

It establishes the existence of bounded continuous functions that are one-to-one almost everywhere in metric spaces with a regular Borel measure.

Findings

Existence of such functions in general metric spaces.

Construction of functions that are injective outside measure-zero sets.

Implications for the structure of continuous functions on measure spaces.

Abstract

It is shown that given a metric space $X$ and a $\sigma$-finite positive regular Borel measure $\mu$ on $X$, there exists a bounded continuous real-valued function on $X$ that is one-to-one on the complement of a set of $\mu$ measure zero.