Definition (Lebesgue-Stieltjes integral)

Let $f:\mathbb{R}^{+}\to \mathbb{R}$ be Borel-measurable, we define the Lebesgue-Stieltjes integral $\int\limits _{\mathbb{R}^{+}}f \, dg$ by $$\int\limits _{\mathbb{R}^{+}}f \, dg=\int\limits _{\mathbb{R}^{+}}f \, d\mu_{g}$$ with the right side being the Lebesgue Integral with the Lebesgue-Stieltjes Measure, $\mu_{g}$.