Exercise 1.7.13 (Lebesgue-Stieltjes measure, absolutely continuous case)

(i)

If $F:R\to R$ is the identity function $F(x)=x$, show that ${\mu }_F$ is equal to Lebesgue measure $m$.

(ii)

If $F:R\to R$ is monotone non-decreasing and absolutely continuous (which in particular implies that $F^{\prime }$ exists and is absolutely integrable, show that ${\mu }_F=m_{F^{\prime }}$ in the sense of Exercise 1.4.48, thus $${\mu }_F(E)={\int \; <!--\; nolimits\; -->}_E{F}^{\prime }(x)\mathit{dx}$$

for any Borel measurable $E$, and

$${\int \; <!--\; nolimits\; -->}_{R}f(x)d{\mu }_F(x)={\int \; <!--\; nolimits\; -->}_{R}f(x)F^{\prime }(x)\mathit{dx}$$

for any unsigned Borel measurable $f:R\to [0,+\infty ]$.