Riemann-Stieltjes Integrals

Recall from calculus that if $f$ is continuous on the interval $[a, b]$, then we define the Riemann integral of $f$ on $[a, b]$ to be:

(1)

\begin{align} \quad \int_a^b f(x) \: dx = \lim_{n \to \infty} \sum_{k=1}^{n} f(x_k^*) \Delta x \end{align}

Where $\Delta x = \frac{b - a}{n}$ and $x_k^* \in [x_{k-1}, x_k]$ for each subinterval $k \in \{1, 2, ..., n \}$. If the limit above exists, we write $\int_a^b f(x) \: dx = A$.

Geometrically, if $f$ is a positive and continuous function on $[a, b]$ then $\int_a^b f(x) \: dx = A$ represented the area underneath the $f$, above the $x$-axis, and bounded by the lines $x = a$ and $x = b$.

We will now look at at much more general type of integral known as Riemann-Stieltjes integrals which we define below.

Definition: Let $I = [a, b]$ be a closed interval and let $f, \alpha$ be functions defined on $[a, b]$. Furthermore, let $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ and for each $k \in \{1, 2, ..., n \}$ let $t_k \in [x_{k-1}, x_k]$ (the $k^{\mathrm{th}}$ subinterval of $[a, b]$ with respect to the partition $P$) and let $\alpha_k = \alpha(x_k) - \alpha(x_{k-1})$. A Riemann-Stieltjes Sum with respect to the partition $P$ and the functions $f$ and $\alpha$ is denoted $\displaystyle{S(P, f, \alpha) = \sum_{k=1}^{n} f(t_k) \Delta \alpha_k}$. The function $f$ is said to be Riemann-Stieltjes Integrable with respect to $\alpha$ on $[a, b]$ if there exists an $A \in \mathbb{R}$ such that for every $\epsilon > 0$ there exists a partition $P_{\epsilon} \in \mathscr{P}[a, b]$ such that for all partitions $P$ finer than $P_{\epsilon}$ ($P_{\epsilon} \subseteq P$) and for any choice of $t_k \in [x_{k-1}, x_{k}]$ we have that $\mid S(P, f, \alpha) - A \mid < \epsilon$. If such an $A \in \mathbb{R}$ exists, we say this Riemann-Stieltjes integral exists and write $\displaystyle{\int_a^b f(x) \: d \alpha(x) = A}$.

Sometimes the notation "$f \in R(\alpha)$ on $[a, b]$" will be used to denote that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$.

If such an $A$ exists, the abbreviated notation $\displaystyle{\int_a^b f \: d\alpha = A}$ can also be used.

Furthermore, the notation $f \in R(\alpha)$ on $[a, b]$ means that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on the interval $[a, b]$.

There are many nice properties of the Riemann-Stieltjes integral that we will look at shortly.