Here's the question I'm hopelessly on:
Let $X_1, X_2, \dots$ be an $iid$ sequence of Bernoulli random variables on $(\Omega,\mathcal{F},\mathbb{P})$ with $\mathbb{P}(X_i = 1)=1/2$. Let
$$X = 3\sum_{k=1}^\infty 4^{-k}X_k.$$
Show that the distribution function $F(x)$ of $X$ is not absolutely continuous w.r.t the lebesgue measure.
I've compared this to the cantor function, which shares many of the same properties. What I'm trying to show is that $F$ only changes values on a set of Lebesgue measure zero, but without success.
Trying to find a set with lebesgue measure zero but positive probability hasn't bourne fruit either.