Laplace–Stieltjes transform

Summary

The Laplace–Stieltjes transform of a function $g$ on $[0, \infty)$ is $\int_0^\infty e^{-s t} \, dg(t)$, a Riemann–Stieltjes integral against $g$ instead of an ordinary integral of $g$. It extends the Laplace transform: when $g$ has a density $f$, it is the Laplace transform of $f$. It also handles jumps of $g$, that is, point masses. In probability, applied to the distribution function of a nonnegative random variable $X$, it gives $\mathbb{E}[e^{-s X}]$.