Laplace transform inversion on the real line is truly ill-conditioned

In this note we consider Laplace transforms of probability distribution functions F(t)F(t) on (0,∞)(0,∞) that have finite integer moments of all orders. We construct a family Fω(t)Fω(t) of distribution functions, whose Laplace transforms differ from that of F(t)F(t) by as little as we want, but such that Fω(t)Fω(t) has a discrete part whereas F(t)F(t) has a density f(t)f(t). Thus we provide one more example of why Laplace transform inversion on the real line is a difficult, ill-conditioned, inverse problem.