Move a function from the integrand into the differential in a Stieltjes-Integral

Question

If I have an integral like this $$\int_{0}^{\infty} e^{-st}f(t)d(\alpha(t)),$$ then is it possible to transform it into a "classic" Laplace-Stieltjes-Integral of the form $$\int_{0}^{\infty}e^{-st}d(\alpha_2(t))?$$

My idea would be to probably calculate $\alpha'(x)$ and then write as the differential the integration of $(\alpha'*f)$.

Answer 1

Well you can't use the same "$\alpha(t)$" in both!
Given two measures, $\alpha(t)$ and $\beta(t)$ then $\int f(t)d\alpha(t)= \int d\beta(t)$ if and only if $d\beta(t)= f(t)d\alpha(t)$ which is the same as saying that the derivative of $\beta$ is f(t) times the derivative of $\alpha$.