Lebesgue-Radon-Nikodym decomposition

Question

Please how to find the Lebesgue decomposition of $\nu$ with respect to the Lebesgue measure $m$ where $\nu$ is the Lebesgue-Stieltjes measure associated to the following function: $$ F(x) = \begin{cases}0 &\text{if }x < 0\,,\\ 2x&\text{if }0 \leq x < 1\,,\\2&\text{if }x \geq 1\,.\end{cases}$$

Thank you very much

Answer 1

If $\nu$ is the one dimensional Lebesgue measure and the $\mu$ is absolutely continuous with respect to $\nu$, then the Radon-Nikodym derivative $\frac{d\mu}{d\nu}$ is just $\frac{dF}{dx}$ where $F(x)=\mu([a,x])$ for $x>a$ and $-\mu([x,a])$ for $x<a$. Here $a$ is any fixed constant. (In the case where $\mu$ is finite it is often convenient to use $(-\infty,x]$ instead.) Try to prove this in general before applying it to this problem.

Answer 2

$d\nu (x)=f(x)dm(x)$ where $f(x)=2$ for $0<x<1$ and $0$ for all other $x$. (The measure is absolutely continuous). [Singular part is $0$].