A measurable version of the Stone-Weierstrass theorem (Q2785037)

Let \(\mu\) be a Borel probability measure on the interval \(I= [0,1]\) and \(\{f_n\}\) be a sequence of bounded \(\mu\)-measurable functions on \(I\) which separate the points of \(I\). The author gives an elementary proof of a Stone-Weierstrass type theorem to the effect that the algebra generated by the \(f_n\) and the constant functions is dense in \(L^p(I,\mu)\) for \(0\leq p<\infty\). He then considers the case where \(\{f_n\}\) does not separate the points of \(I\) and reformulates this result in terms of measurable partitions of \(I\) and closed subalgebras of \(L^0(I,\mu)\) which contain the constant functions.