📂Analysis

Riemann-Stieltjes Integral

Overview

The Riemann-Stieltjes integral is a generalization of the Riemann integral, sometimes simply referred to as Stieltjes integral. The Riemann integral is a special case of the Riemann-Stieltjes integral where $\alpha (x)=x$.

The process of defining the Riemann-Stieltjes integral is the same as the process of defining the Riemann integral, so details on the notation and buildup are omitted here.

Definition

Let $\alpha : [a,b] \to \mathbb{R}$ be a monotonically increasing function, and let $\Delta \alpha_{i}=\alpha (x_{i})-\alpha (x_{i-1})$. Then, since $\alpha$ is a monotonically increasing function, $\Delta \alpha_{i} \ge 0$ holds.

For a bounded function $f : [a,b] \to \mathbb{R}$ and a partition $P$ of $[a,b]$, define $U, L$ as follows.

$$\begin{align} U(P,f,\alpha) &:= \sum \limits _{i=1} ^n M_{i} \Delta \alpha_{i} \\ L(P,f,\alpha) &:= \sum \limits_{i=1} ^n m_{i} \Delta \alpha_{i} \end{align}$$

Define $(1), (2)$ as the upper and lower Riemann-Stieltjes sums of $f$ for $\alpha$ in $[a,b]$.

Taking the $\inf, \sup$ over all possible partitions $P$ of interval $[a,b]$ for $(1), (2)$ gives us the upper and lower Riemann-Stieltjes integrals of $f$ for $\alpha$ in $[a,b]$.

$$\begin{align*} \overline {\int _{a} ^b} f d\alpha &:= \inf\limits_{P} U(P,f,\alpha) \\ \underline {\int _{a} ^b} f d\alpha &:= \sup\limits_{P} L(P,f,\alpha) \end{align*}$$

If the upper and lower integrals are equal, it is referred to as the Riemann-Stieltjes integral of $f$ for $\alpha$ in $[a,b]$ and is denoted as follows.

$$\int _{a} ^b f d\alpha = \int _{a}^b f(x) d\alpha (x) = \overline {\int _{a} ^b} f d\alpha = \underline {\int _{a} ^b} f d\alpha$$

If the Stieltjes integral of $f$ exists, then $f$ is Riemann-Stieltjes integrable for $\alpha$ in $[a,b]$, denoted as:

$$f \in \mathscr{R}(\alpha) = \left\{ f : f \text{ is Riemann-Stieltjes integrable} \right\}$$