Given two (sufficiently good) single-variable functions
\begin{equation}f, g: [a,b] \mapsto \mathbb{R}, \text{ here } a,b \in \mathbb{R}\end{equation}
the Riemann–Stieltjes integral is defined as
\begin{equation} \int_{a}^{b} f \,dg = \lim_{N\to\infty} \sum_{i=1}^N f\left(a+i\Delta_N\right)\left[ g\left(a+i\Delta_N\right) - g\left(a+(i-1)\Delta_N\right) \right], \\\quad \text{here } \Delta_N=\frac{b-a}{N} \end{equation}
I was wondering if there is a Riemann-Stieltjes integral definition for multivariate case, e.g. how to define $\int_S f \, dg$ for multivariate functions, e.g. when both $f, g: S \mapsto \mathbb{R}$, where $S \subset \mathbb{R}^n $ ? (We can assume $S$ is a hyper-rectangle for simplicity.)
This question has arisen from the problem of how to calculate the mean of a function of random vector. E.g., having a random vector $X: \Omega \mapsto \mathbb{R}^n$ with an arbitrary cdf $F_X$ and given a function $g: \mathbb{R}^n \mapsto \mathbb{R}$, how to numerically approximate $\mathbb{E}[g(X)] = \int_{\mathbb{R}^n} g(x) \, dF_X(x)$?
In the book Hildebrandt Theophil Henry Introduction to the theory of integration-(1963) there is Riemann-Stieltjes for 2 variables, started from page 123.
I am not copying definition here as it is in book, but am ready to discuss some particular moment, if any.
Addition. Accordingly conversation in chat I am adding definition of Riemann-Stieltjes here directly for case $f:\mathbb{R}^n \to \mathbb{R}$. It can be done in several ways and first classical one is consider step functions. We take $F:\mathbb{R}^n \to \mathbb{R}$ increasing with respect to any variable and step function $h$, piecewise constant on rectangle $I=[a_1,b_1] \times\cdots \times [a_n,b_n]$ and define $|F(I)|=\Delta_1\cdots \Delta_n F(I)$, $\Delta_j F(I)= F(x_1, \cdots,x_{j-1},b,x_{j+1},\cdots,x_n)-F(x_1, \cdots,x_{j-1},a,x_{j+1},\cdots,x_n)$. We define integral for $h$ with respect to $F$ as $$\int\limits_{J}h(x)dF(x)=\sum\limits_{i=1}^{n}c_i|F(I_i)|$$ where $J=\cup_{i=1}^{n} I_i$ .
Now any $f$ is Riemann-Stieltjes integrable with respect to $F$ when for $\forall \epsilon >0$ exists step functions $h_1, h_2$ such that $h_1 \leqslant f \leqslant h_2$ and $$\int\limits_{J}h_2(x)dF(x) - \int\limits_{J}h_1(x)dF(x) < \epsilon$$ and Riemann-Stieltjes integral for $f$ is defined as $$\int\limits_{J}f(x)dF(x) = \sup \left\{ \int\limits_{J}h(x)dF(x): h \leqslant f,\ h\ \text{step function} \right\}$$
Second possibility is to define integral as limit of Riemann-Stieltjes sum $$\int\limits_{J}f(x)dF(x) =\lim\limits_{\max |I_i| \to 0}\sum_{i=1}^{n}f(\xi_i)|F(I_i)|$$ where $\xi \in I_i$.