I was thinking about this and I could not get to an answer. To illustrate my point, assume I have a random variable $X:\Omega\rightarrow\mathbb{R}$ and a measure $\mu$ and I want to compute its expectation:
$$EX=\int X d\mu$$
Can I, WITHOUT loss of generality, assume that I can represent this integral as a Lebesgue-Stieltjes integral?
Provided that $\Omega = \mathbb{R}$ and $\mu$ assigns finite measure to bounded sets, the answer is yes. Set $$G(x) = \begin{cases} \mu([0,x]), & x \ge 0 \\ -\mu((x,0)), & x < 0. \end{cases}$$ Then you can check that $G$ is nondecreasing and right continuous (use countable additivity), and for each bounded interval $(a,b]$ we have $\mu((a,b]) = G(b) - G(a)$. You can also verify that $\int X\,d\mu = \int X\,dG$.