Wikipedia gives that the moment-generating function for a real random variable $X$ with cumulative distribution function $F$ equals
$M_x(t) = \int \limits_{-\infty}^\infty e^{tx}\, dF(x)$, using the Riemann-Stieltjes integral.
It defines the (bilateral) Laplace-Stieltjes transform of a real function $g(x)$ to be
a Lebesgue-Stieltjes integral of the form $\{\mathcal{L}^*g\}(s) = \int \limits_{-\infty}^\infty e^{-sx} dg(x)$.
These two definitions look identical to me, except for the sign of the argument and the slightly different type of integral. Is the choice of Riemann-Stieltjes vs. Lebesgue-Stieltjes important here? Are there any CDFs for which these two quantities disagree (beyond the opposite sign of the argument)?