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.?
Best Answer
Variants of the above Lebesgue--Stieltjes partial integral results are given in Theorem 21.67(iv) (it holds for general BV functions, by Remarks 21.68) of Real and Abstract Analysis: A modern treatment of the theory of functions of ... E. Hewitt, K. Stromberg p. 419
They seem to answer your question, though at the cost of having terms like $(F(x+)+F(x-))/2$ in place of $F(x)$.