Question

This exercise deals with functions F that are absolutely continuous on [a; b]
and are increasing. Let A = F(a) and B = F(b).
(a) There exists such an F that is in addition strictly increasing, but such that
F'(x) = 0 on a set of positive measure.
The F in (a) can be chosen so that there is a measurable subset E is subset of [A;B],
m(E) = 0, so that F^-1(E) is not measurable.