Question: Prove that if $f$ is continuous in [a,b] and $g$ is monotone increasing and continuous from the right, then \begin{equation} \int_a^bf(x)dg(x)=\int_{(a,b]}f(x)dg(x), \end{equation} where $dg=d\mu_g$, and $\mu_g$ is the Lebesgue-Stieltjes measure induced by $g$.
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I have no idea where to start the proof. Hope get an answer here.Thanks.