Let $f(x)$ be the Cantor-Lebesgue function. Write $$g(x)=x+f(x)$$ and let $\mathbb{B}$ be the sigma algebra generated by preimages of intervals under $g$, $\mathbb{C}$ be the sigma algebra which is the intersection of the algebra of Lebesgue measurable sets and $\mathbb{B}$, $\mu$ be the restriction of Lebesgue measure to $\mathbb{C}$, and define measure $v$ on $\mathbb{C}$ by $$v(A)=m(g(A))$$ where m is the lebesgue measure on [0,2].
Question: Find the Lebesgue decomposition of $v$ with respect to $\mu$. What is the Radon-Nikodym derivative of the absolute continuous part of $v$ ?
I know that $g(x)$ is a strictly increasing function. But I think that in order to solve this problem, we need to find more about the structure of sigma algebra $\mathbb{C}$. I don't know how to approach it.