Let $\mu$ be the Lebesgue-Stieltjes measure on $\mathbb{R}$ corresponding to the distribution function, $F$ where
$$F(x) = \left\lbrace \begin{array}{ll} 0& \text{if} \,\, x<0\\ x+1& \text{if} \,\, 0\leq x<1\\ 2x+3& \text{if} \,\, 1\leq x<2\\ 8& \text{if} \,\, x\leq 2 \end{array} \right. $$ Then Find the absolutely continuous part and the singular part of the measure with respect to Lebesgue measure on $\mathbb{R}$.
I know every sigma finite measure can be composite into two parts but in this question. I do not know how to begin. Please help me out.