This question comes from my attempts at understanding an example presented by Bill Gasarch on his blog. The example is of a continuous strictly increasing function whose derivative is zero almost everywhere. The example is apparently discussed in the book Probability and Measure by Patrick Billingsley, but I currently do not have access to it.
Gasarch explains the background very well. Given a parameter $p \in (0,1)$, he describes a continuous function $F:[0,1]\to[0,1]$ which is increasing, $F(0) = 0$, $F(1) = 1$, but such that $F' = 0$ a.e. The derivative $f = F'$, which is only defined almost everywhere, must be nonnegative and should satisfy the following functional equation $f(x) = \begin{cases} 2pf(2x) & \text{when $x \in (0,1/2)$,} \\ 2(1-p)f(2x-1) & \text{when $x \in (1/2,1)$,} \end{cases}$ almost everywhere. It would seem, from the claimed example, that a nonnegative measurable function that satisfies this functional equation almost everywhere must be $0$ almost everywhere. However, this is not true since every constant function satisfies this functional equation when $p = 1/2$. Thinking about the dynamic properties of the transformation $T(x) = \begin{cases} 2x & \text{when $x \in [0,1/2)$,} \\ 1/2 & \text{when $x = 1/2$ (say),} \\ 2x-1 & \text{when $x \in (1/2,1]$,} \end{cases}$ it does seem that the space of solutions to the above equation is heavily constrained.
Is there a nice characterization of the space of solutions to the above functional equation? A general characterization would be best but a characterization for special cases (e.g. $f \geq 0$, $p = 1/2$) would be welcome.