# Flexible versions of the Stone-Weierstrass Theorem in General and Applications to Probability Theory

## TL;DR

This paper develops two adaptable versions of the Stone-Weierstrass theorem to address issues in probability theory when the classical assumptions are not met, enhancing its applicability.

## Contribution

It introduces flexible variants of the Stone-Weierstrass theorem tailored for probability theory, overcoming limitations of the classical version.

## Key findings

- Two new versions of the Stone-Weierstrass theorem are proposed.
- The flexible versions are easier to apply in probability contexts.
- The methods extend the theorem's applicability to non-separated compact sets.

## Abstract

When applying the classical Stone-Weierstrass common version in Probability Theory for example, and in other fields as well, problems may arise if all points of the compact set are not separated. A solution may consist in going back to the proof and finding alternative versions. In this note, we did it and come back with two flexible versions which are easily used for the needs of classical Probability Theory.