The Comparison Theorem for R-S Integrals with Increasing Integrators.

# The Comparison Theorem for Riemann Stieltjes Integrals with Increasing Integrators

Suppose that $f$ and $g$ are functions defined on $[a, b]$ and that $\alpha$ is an increasing function on $[a, b]$. By looking at any Riemann-Stieltjes sum corresponding to any partition $P \in \mathscr{P}[a, b]$ we see that the value of the Riemann-Stieltjes sums more so depend on the functions $f$ and $g$.

If $f(x) \geq g(x)$ for all $x \in [a, b]$ then it is reasonable to expect that for any partition $P \in \mathscr{P}[a, b]$ that $S(P, f, \alpha) \leq S(P, g, \alpha)$. If $f$ and $g$ are Riemann-Stieltjes integrable, then more can be said about the Riemann-Stieltjes integrals of $f$ and $g$ with respect to $\alpha$ on $[a ,b]$ which we look at in the following theorem.

 Theorem 1: Let $f$ and $g$ be functions defined on $[a, b]$ and let $\alpha$ be an increasing function on $[a, b]$. If $f$ and $g$ are both Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ and $f(x) \leq g(x)$ for all $x \in [a, b]$ then $\displaystyle{\int_a^b f(x) \: d \alpha (x) \leq \int_a^b g(x) \: d \alpha (x)}$.
• Proof: Let $\alpha$ be an increasing function on $[a, b]$ and suppose that $f$ and $g$ are both Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ and $f(x) \leq g(x)$ for all $x \in [a, b]$. Let $P = \{ a =x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ be any partition on $[a, b]$.
• Since $\alpha$ is increasing on $[a, b]$ we have that $\Delta \alpha_k = \alpha(x_k) - \alpha(x_{k-1}) \geq 0$ for all $k \in \{1, 2, ..., n \}$, and since $f(x) \geq g(x)$ for all $x \in [a, b]$ we see that $f(t_k) \leq g(t_k)$ for each $t_k \in [x_{k-1}, x_k]$ for $k \in \{1, 2, ..., n \}$. So for every partition $P \in \mathscr{P}[a, b]$ we have that:
(1)
\begin{align} \quad f(t_k) \Delta \alpha_k & \leq g(t_k) \Delta \alpha_k \\ \quad \sum_{k=1}^{n} f(t_k) \Delta \alpha_k & \leq \sum_{k=1}^{n} g(t_k) \Delta \alpha_k \\ \quad S(P, f, \alpha) & \leq S(P, g, \alpha) \end{align}
• Since $f$ and $g$ are Riemann-Stieltjes integrable, we see that as $\| P \| \to 0$ we have that $S(P, f, \alpha) \to \int_a^b f(x) \: d \alpha (x)$ and $S(P, g, \alpha) \to \int_a^b g(x) \: d \alpha (x)$, and from the inequality above, we get:
(2)
\begin{align} \quad \int_a^b f(x) \: d \alpha (x) \leq \int_a^b g(x) \: d \alpha (x) \quad \blacksquare \end{align}