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

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.