Riemann Stieltjes-Integrals with Constant Integrators

Recall from the Riemann Stieltjes-Integrals with Constant Integrands page that if $f(x) = C$ for some $C \in \mathbb{R}$ and $\alpha$ is a function defined on $[a, b]$ then $f$ is always Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ and furthermore:

(1)

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

We will now see that if instead $f$ is any function defined on $[a, b]$ and the integrator $\alpha (x) = C$ for some $C \in \mathbb{R}$ then $f$ is always Riemann-Stieltjes integrable with respect to $\alpha$ and the Riemann-Stieltjes integral will always equal $0$.

Theorem 1: Let $f$ be any function defined on $[a, b]$ and let $\alpha (x) = C$ for some $C \in \mathbb{R}$. Then $f$ is Riemann-Stieltjes Integrable with respect to $\alpha$ on $[a, b]$ and $\int_a^b f(x) \: d C = 0$.

• Proof: Let $f$ be any function defined on $[a, b]$ and let $\alpha(x) = C$ for some $C \in \mathbb{R}$. Also 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 $t_k \in [x_{k-1}, x_k]$ and noting that $\alpha(x_k) = C$ for all $k \in \{0, 1, 2, ..., n \}$ we have that:

(2)

\begin{align} \quad S(P, f, C) &= \sum_{k=1}^{n} f(t_k) \Delta \alpha_k \\ \quad S(P, f, C) &= \sum_{k=1}^{n} f(t_k) [\alpha(x_k) - \alpha(x_{k-1})] \\ \quad S(P, f, C) &= \sum_{k=1}^{n} f(t_k) [C - C] \\ \quad S(P, f, C) &= \sum_{k=1}^{n} f(t_k) \cdot 0 \\ \quad S(P, f, C) &= 0 \end{align}

• Notice that the Riemann-Stieltjes sum above is independent of the choice of partition $P$ and the $t_k$'s. So if $A = 0$ then for all $\epsilon > 0$ we have that there exists a partition $P_{\epsilon} \in \mathscr{P}[a, b]$ (any partition) such that if $P$ is finer than $P_{\epsilon}$, i.e., $P_{\epsilon} \subseteq P$ then:

(3)

\begin{align} \quad 0 = \mid S(P, f, C) - 0 \mid < \epsilon \end{align}

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

(4)

\begin{align} \quad \int_a^b f(x) \: dC = 0 \quad \blacksquare \end{align}