Does the set of points where Lebesgue Stieljes distribution function have zero derivative have zero $ \mu$ measure?

Question

By distribution function I mean a right continuous increasing function on $\mathbb{R}$ and $\mu(a,b)= F(b)-F(a)$ for every interval $(a,b)$ as defined in Folland and other books. Now F is increasing hence its derivative exists Lebesgue almost everywhere. Question, does this also apply to all Lebesgue Steiljes measures? Does F' exist for $\mu$ a.e. x as well? Finally, if it does, then is F'(x)>0 $\mu$ a.e. x? Thank you for any insight.