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Riemann-Stieltjes integration is available when a integrator is monotonically increasing. However, three texts i have, all define Lebesgue-Stieltjes measure with respect to monotonically increasing right-continuous function.

I don't understand why these texts restrict such hypotheses.

Let $F:\mathbb{R}\rightarrow \mathbb{R}$ be a monotonically increasing function and define $G(x)=F(x^+)$. Then $G$ is monotonically increasing right continuous. Then we can define Lebesgue-stieltjes measure $\mu_G$associated with $G$. ($\mu_G((a,b])=G(b)-G(a)$).

Is it possible to construct a measure $\phi$ such that $\phi((a,b))=\phi(b^-) - \phi(a+)$, using $\mu_G$?

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I think the reason is because there is a sort of converse. That is let $\mu$ be a Borel measure on the reals (which is bounded on all bounded Borel sets). Then the function $F$ defined by $$F(x)=\begin{cases} \mu((0,x]) &\text{if }x>0 \\ 0 &\text{if} x=0\\ -\mu((x,0])&\text{if }x<0 \end{cases}$$

Is a monotone, right-continuous function such that $\mu=\mu_F$, and this function is unique by a constant.

I'm not sure what happens when you allow $\mu$ to take infinite values on bounded sets.

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