Lebesgue-Stieltjes measure (${\mu }_F$)

An increasing right-continuous function $F:\mathbb{R}\to \mathbb{R}$ determines a measure byThe Radon-Nikodym theorem and Lebesgue differentiation theorem identify the density of the absolutely continuous part of this measure with $F^{\prime }$ almost everywhere.

The Lebesgue–Stieltjes integral $\int fdF$ integrates $f$ against the Lebesgue-Stieltjes measure generated by an increasing right-continuous function $F$. An atom at $x$ contributes $f(x)\Delta F(x)$.