I am concerned with the following problem:
I am wondering if there exists any sort of integration by parts formula for a multidimensional Lebesgue-Stieltjes integral. In my case the integral is given by (1) and its domain is a multidimensional subset of $\mathbb{R}^n$. I suspect that this would involve the function $f$ instead of the corresponding measure $\mu$ according to Theorem 2 below.
I am looking for a result similar to the well known result in the case of $n=1$, i.e.: $$\int\limits_a^b g(t) \, df(t) = g(b)f(b)-g(a)f(a)-\int\limits_a^b f(t) dg(t)$$
I'd be greatful for any hint on literature that contains relevant information on such multidimensional scenarios.
So far I found plenty of literature on the one-dimensional case, i.e. $\Omega \subset \mathbb{R}$, but nothing definitive on $\Omega \subset \mathbb{R}^n$ for $n>1$.
Let's consider the following well known result from duality theory in Analysis:
Theorem 1(cf. Aliprantis Border, Infinite-Dimensional Analysis Theorem 14.14)
Let $\Omega \subset \mathbb{R}^n$ and $\Phi: C_c(\Omega) \rightarrow \mathbb{R}$ be a continuous linear functional, where $C_c(\Omega)$ is the set of all real valued continous functions on $\Omega$ with compact support. Then $\Phi$ can be expressed the follwing way:
$$\Phi(g) = \int\limits_{\Omega} g(x) \, d\mu(x) \mspace{1in} \forall g \in C_c(\Omega)\tag{1}$$
where $\mu$ is a regular signed Borel measures of bounded variation, which is uniquely determined.
To further work with this measure $\mu$ I found the following interesting fact:
Let's define $$\Delta_h f(x) := \sum\limits_\delta (-1)^{\sum_{i=1}^n \delta_i} f(x-h(\delta)) \tag{2}$$ with $\delta_i$ being either $0$ or $1$, $h = (h_1,\dots,h_n)$ and $h(\delta) = (h_1 \delta_1, \dots, h_n \delta_n)$ and the sum over $delta$ meaning the sum over all possible binary vectors with length $n$. Then the following theorem holds
Theorem 2 (cf. Aliprantis Border, Infinite-Dimensional Analysis Theorem 10.50)
If $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is continuous from above and satisfies $\Delta_h f(x) \geq 0$ for all $x \in \mathbb{R}^n$ and all $h \in \mathbb{R}_+^n$, then there exists a unique Borel measure $\mu$ on $\mathbb{R}^n$ satisfying (2).
Conversely, if $\mu$ is Borel measure on $\mathbb{R}^n$, then there exists a function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ (unique up to translation) that is continuous from above, satisfies $\Delta_h f(x) \geq 0$ for all $x \in \mathbb{R}^n$ and all $h \in \mathbb{R}_+^n$, and satisfies (2).
