Let $F$ be a probability distribution function on $\mathbb{R}$ whose support is contained in $[0,1]$. Is the following integration by parts hold for arbitrary such $F$: $$\int_{[0,a]}x \mathrm{d}F(x)= aF(a)-\int_0^aF(x)\mathrm{d}x?$$
Can anyone give me some classic references on the integration by parts for Lebesgue-Stieltjes integration? Thank you!