Let $f: \mathbb{R} \to \mathbb{R}$ be a Borel measurable and non-negative function; let $g: \mathbb{R} \to \mathbb{R}$ have bounded variation but not necessarily be right-continuous. Then, the Lebesgue Stieltjes integral $$I(t) := \int_{(0,t]} f(u) \mathrm{d}g(u)$$ is well defined for $t>0$.
Assume that $f,g$ are such that the integral $I$ has bounded variation. Let $h: \mathbb{R} \to \mathbb{R}$ be a Borel measurable and non-negative function. My question: under which additional conditions on $f,g,h$ does the following identity hold $$\int_{(0,t]} h(u) \mathrm{d} I(u) = \int_{(0,t]} h(u) f(u) \mathrm{d}g(u)?$$ Any help would be appreciated, including naming this identity so that I can better search for information. I have not made any progress on the problem.