The notation$\int_0^t\vert f\vert\,\vert dg\vert$ represents the Lebesgue-Stieltjes integral of the function $\vert f \vert$ with respect to the total variation of the function $ g $.
Suppose that the function $f$ satisfies $\int_0^t\vert f \vert\,\vert dg\vert<\infty $, so it is integrable with respect to $g$ in the Lebesgue-Stieltjes sense. If $\vert f_n\vert\le\vert f \vert$ is a sequence of bounded functions tending to a limit function $\alpha$ then how can we conclude that
$$\int_0^t f_n\,dg\rightarrow\int_0^t\alpha\,dg $$
Is there some sort of dominated convergence for signed measures ?