Riemann-Stieltjes Integrals with Constant Integrands

We will now look at a specific type of integrand that is Riemann-Stieltjes integrable with respect to any integrator $\alpha$. That special type of integrand is the collection of constant functions $f(x) = C$ where $C \in \mathbb{R}$.

Theorem 1: Let $f(x) = C$ be a constant function on $[a, b]$ where $C \in \mathbb{R}$ and let $\alpha$ be any function on $[a, b]$. Then $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ and $\displaystyle{\int_a^b C \: d \alpha (x) = C[\alpha(b) - \alpha(a)]}$.

• Proof: Let $f(x) = C$ on $[a, b]$ where $C \in \mathbb{R}$ and let $\alpha$ be any function on $[a, b]$. Let $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ by any partition and consider the Riemann-Stieltjes sum for any choices of $t_k \in [x_{k-1}, x_k]$:

(1)

\begin{align} \quad S(P, C, \alpha) &= \sum_{k=1}^{n} C \Delta \alpha_k \\ \quad S(P, C, \alpha) &= C \sum_{k=1}^{n} \Delta \alpha_k \\ \quad S(P, C, \alpha) &= C \sum_{k=1}^{n} [\alpha(x_{k}) - \alpha(x_{k-1})] \\ \quad S(P, C, \alpha) &= C [\alpha(x_n) - \alpha(x_0)] \\ \quad S(P, C, \alpha) &= C[\alpha(b) - \alpha(a)] \end{align}

• Notice that the Riemann-Stieltjes sum above is independent of the partition $P$ and the choices of $t_k \in [x_{k-1}, x_k]$. So if $A = C[\alpha(b) - \alpha(a)]$ then for all $\epsilon > 0$ there exists a partition $P_{\epsilon} \in \mathscr{P}[a, b]$ (any partition) such that if $P$ is finer than $P_{\epsilon}$ ($P_{\epsilon} \subseteq P$) then:

(2)

\begin{align} \quad 0 = \mid S(P, C, \alpha) - C[\alpha(b) - \alpha(a)] \mid < \epsilon \end{align}

• Therefore $f(x) = C$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ and furthermore:

(3)

\begin{align} \quad \int_a^b C \: d \alpha (x) = C[\alpha(b) - \alpha(a)] \quad \blacksquare \end{align}