Lebesgue–Stieltjes integration

Summary

Lebesgue–Stieltjes integration integrates a function $f$ against a non-decreasing function $g$ on the real line, written $\int f \, dg$. It is the Lebesgue integral with respect to the Lebesgue–Stieltjes measure of $g$, which gives each interval $(a, b]$ the mass $g(b) - g(a)$. It extends the Riemann–Stieltjes integral to all measurable functions, and to functions $g$ of bounded variation by splitting $g$ into monotone parts. It is the language of probability distributions on the line, where $g$ is a cumulative distribution function.