---
tag: D9H2AGW198
kind: definition
name: Laplace–Stieltjes transform
msc: [44A10, 26A42, "60E10"]
defines: KHRTBP37VD
uses: [F34S6P61SW, 8NMH42GJRK, 05XT5AVJBK]
sources:
  - url: https://en.wikipedia.org/wiki/Laplace%E2%80%93Stieltjes_transform
---

## Definition

Let $g \colon [0, \infty) \to \C$ be a function of [[bounded variation|F34S6P61SW]] on each interval $[0, T]$ with $T > 0$.
The *Laplace–Stieltjes transform* of $g$ is the function
$$\mathcal{L}^*\{g\}(s) = \int_0^\infty e^{-s t} \, dg(t) = \lim_{T \to \infty} \int_0^T e^{-s t} \, dg(t),$$
defined for each complex $s$ for which the limit exists.
Each integral $\int_0^T e^{-s t} \, dg(t)$ is a [[Riemann–Stieltjes integral|8NMH42GJRK]].

## Remarks

Example: let $a > 0$ and $g(t) = 0$ for $t < a$, $g(t) = 1$ for $t \ge a$.
Then $\int_0^T e^{-s t} \, dg(t) = e^{-a s}$ for every $T > a$, so $\mathcal{L}^*\{g\}(s) = e^{-a s}$ for every complex $s$.
If $g(t) = \int_0^t f(\tau) \, d\tau$ with $f$ continuous, then $dg = f(t) \, dt$ and $\mathcal{L}^*\{g\}$ is the [[Laplace transform|05XT5AVJBK]] of $f$.
Non-example: $g(t) = e^{t^2}$ is of bounded variation on each $[0, T]$, but $\int_0^T e^{-s t} \, dg(t) = \int_0^T 2 t e^{t^2 - s t} \, dt$ diverges as $T \to \infty$ for every $s$, so no transform exists at any point.
In probability one often uses the Lebesgue–Stieltjes form $\int_{[0, \infty)} e^{-s t} \, dG(t)$ for a distribution function $G$, which counts a possible jump at $t = 0$.
With $G(t) = 0$ for $t < 0$ it equals $\mathbb{E}[e^{-s X}]$ for $X \ge 0$ with distribution function $G$.
If $g(0) = 0$ and $\abs{g(t)} \le M e^{a t}$ for all $t \ge 0$, integration by parts gives $\mathcal{L}^*\{g\}(s) = s \, \mathcal{L}\{g\}(s)$ for $\operatorname{Re} s > a$.