# Riemann–Stieltjes Integral for Multivariate Functions

Given two (sufficiently good) single-variable functions

$$$$f, g: [a,b] \mapsto \mathbb{R}, \text{ here } a,b \in \mathbb{R}$$$$

the Riemann–Stieltjes integral is defined as

$$$$\int_{a}^{b} f \,dg = \lim_{N\to\infty} \sum_{i=1}^N f\left(a+i\Delta_N\right)\left[ g\left(a+i\Delta_N\right) - g\left(a+(i-1)\Delta_N\right) \right], \\\quad \text{here } \Delta_N=\frac{b-a}{N}$$$$

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 $$\mathbb{E}[g(X)] = \int_{\mathbb{R}^n} g(x) \, dF_X(x)$$?

• Multivariate functions or vector functions? Jul 30, 2020 at 13:36
• Is it different? Jul 30, 2020 at 14:13
• Essentially depends on definition - which one you are interesting in: $\mathbb{R}^n \to \mathbb{R}$ or $\mathbb{R}^n \to \mathbb{R}^m$ or some other? Jul 30, 2020 at 15:28
• It is $\mathbb{R}^n \mapsto \mathbb{R}$ primarily, but $\mathbb{R}^n \mapsto \mathbb{R}^m$ is also interesting Jul 31, 2020 at 7:15
• @zkutch, updated my question based on your comment Jul 31, 2020 at 8:05

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 $$\xi \in I_i$$.