Define $F(x) = \begin{cases} 0 & x < -1 \\ x + 2 & -1\leq x < 0 \\ x^2 + 3 & 0\leq x < 1\\ 6 & x \ge 1 \end{cases}$
I want to find the Lebesgue decomposition of $\mu_F$ (w.r.t Lebesgue measure), the Lebesgue-Stieljes measure associated with the function $F$, and then find the Radon-Nikodym derivative of the absolutely continuous part of the decomposition.
I note that $\mu_{F1 + F2} = \mu_{F1} + \mu_{F2}$, so I try to decompose $F$ itself as the sum of a continuous function and something else:
$F = G + H$ where
$G(x) = \begin{cases} 0 & x < -1 \\ x + 1 & -1\leq x < 0 \\ x^2 + 1 & 0\leq x < 1\\ 2 & x \ge 1 \end{cases}$
$H(x) = \begin{cases} 0 & x < -1 \\ 1 & -1\leq x < 0 \\ 2 & 0\leq x < 1\\ 4 & x \ge 1 \end{cases}$
My question is then what would $\mu_H$ be? My intuition tells me it is equal to $\delta_{-1} + 2\delta_{0} + 4\delta_{1}$, but I am not able to prove this. If I have this, I can then prove $\mu_H, \mu$ are mutually singular by decomposing into $\{-1, 0, 1\}$ and the rest of $\mathbb{R}$. Also, I am not sure how to find the Radon-Nikodym derivative of $\mu_G$. I think I can just take the regular derivative of $F$, but I am unsure as to why it is the same thing.