Let $g$ be a continuous (not need to be monotone) function of bounded variation and $f$ be a measurable function. $m$ is Lebesgue measure on $\mathbb{R}$, and $m_g$ is the signed Lebesgue-Stieltjes measure that $m_g((a,b]) = g(b) - g(a)$.
Question: Does $\int_a^{b}f(g(x))dm_g = \int_{g(a)}^{g(b)}f(x)dm$ hold when the integral of either side exists?
I can prove the above equality when g is monotone using the change of variable formula for positive measures in this post: Is there a change of variables formula for a measure theoretic integral that does not use the Lebesgue measure.
