quality:corruption, information:extraction
Laplace–Stieltjes transform

The (unilateral) Laplace–Stieltjes transform of a function g : ℝ → ℝ is defined by the Lebesgue–Stieltjes integral

{ L ∗ g } (s) = ∫ 0 ∞ e − s t d g (t). {\displaystyle \{{\mathcal {L}}^{*}g\}(s)=\int _{0}^{\infty }e^{-st}\,d\,g(t)~.}

The function g is assumed to be of bounded variation. If g is the antiderivative of f:

g (x) = ∫ 0 x f (t) d t {\displaystyle g(x)=\int _{0}^{x}f(t)\,d\,t}

then the Laplace–Stieltjes transform of g and the Laplace transform of f coincide. In general, the Laplace–Stieltjes transform is the Laplace transform of the Stieltjes measure associated to g. So in practice, the only distinction between the two transforms is that the Laplace transform is thought of as operating on the density function of the measure, whereas the Laplace–Stieltjes transform is thought of as operating on its cumulative distribution function.