I want to use Integration by parts for general Lebesgue-Stieltjes integrals. The following theorem can be found in the literature:
Theorem: If $F$ and $G$ are right-continuous and non-decreasing functions, we have that: $$ \int_{(a,b]}G(x)\text{d}F(x)=F(b)G(b)-F(a)G(a)- \int_{(a,b]}F(x-)\text{d}G(x),$$ where $F(x-)$ is the left limit of $F$ in $x$.
Does the following result hold:
Theorem: If $F$ and $G$ are left-continuous and non-decreasing functions, we have that: $$ \int_{[a,b)}G(x)\text{d}F(x)=F(b)G(b)-F(a)G(a)- \int_{[a,b)}F(x+)\text{d}G(x),$$ where $F(x+)$ is the right limit of $F$ in $x$.
Is it possible to combine these result. So use integration by parts when $F$ is right cont., $G$ is left cont.?
You can Compare with the following theorem,
Theorem: Suppose $f$ and $g$ are bounded functions with no common discontinuities on the interval $[a,b]$, and the Riemann-Stieltjes integral of $f$ with respect to $g$ exists. Then the Riemann-Stieltjes integral of $g$ with respect to $f$ exists, and
$$\int_{a}^{b} g(x)df(x) = f(b)g(b)-f(a)g(a)-\int_{a}^{b} f(x)dg(x)\,. $$