Let $F : \mathbb R\to\mathbb R$ be a distribution function (increasing right continuous), and $m_F : M_F\to[0,\infty]$ the Lebesgue-Stieltjes measure generated by $F$.
In most cases I've come across there are sets in $M_F$ that are not Borel sets.
Is the general statement "For every choice of $F$, $M_F$ contains a set which isn't a Borel set" a theorem?