Riemann's Condition Part 2

# Riemann's Condition Part 2 - The Existence of Riemann-Stieltjes Integrals with Increasing Integrators

On the Riemann's Condition Part 1 - The Existence of Riemann-Stieltjes Integrals with Increasing Integrators page we proved that $a) \implies b)$ in the Theorem below. We will now finish the proof showing the significance of Riemann's condition.

 Theorem 1 (Riemann's Condition): Let $f$ be a function defined on $[a, b]$ and let $\alpha$ be an increasing function on $[a, b]$. Then the following statements are equivalent: a) $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$. b) For every $\epsilon > 0$ there exists a partition $P_{\epsilon}$ such that if $P$ is finer then $P_{\epsilon}$ ($P_{\epsilon} \subseteq P$) then $0 \leq U(P, f, \alpha) - L(P, f, \alpha) < \epsilon$ (Riemann's Condition). c) $\displaystyle{\overline{\int_a^b} f(x) \: d \alpha(x) = \underline{\int_a^b} f(x) \: d \alpha (x)}$.
• Proof of $b) \implies c)$: Suppose that Riemann's condition holds. The for all $\epsilon > 0$ there exists a partition $P_{\epsilon} \in \mathscr{P}[a, b]$ such that for all partitions $P$ finer than $P_{\epsilon}$ we have that:
(1)
\begin{align} \quad U(P, f, \alpha) - L(P, f, \alpha) < \epsilon \\ \quad U(P, f, \alpha) < L(P, f, \alpha) + \epsilon \end{align}
• Therefore we have that:
(2)
\begin{align} \overline{\int_a^b} f(x) \: d \alpha (x) \leq U(P, f, \alpha) < L(P, f, \alpha) + \epsilon \leq \underline{\int_a^b} f(x) \: d \alpha (x) + \epsilon \end{align}
• Hence we have that $\overline{\int_a^b} f(x) \: d \alpha(x) \leq \underline{\int_a^b} f(x) \: d \alpha (x)$. But we always have that $\underline{\int_a^b} f(x) \: d \alpha (x) \leq \overline{\int_a^b} f(x) \: d \alpha (x)$, so from these two inequalities we get that:
(3)
\begin{align} \quad \overline{\int_a^b} f(x) \: d \alpha (x) = \underline{\int_a^b} f(x) \: d \alpha (x) \end{align}
• Proof of $c) \implies a)$: Suppose that $\overline{\int_a^b} f(x) \: d \alpha (x) = \underline{\int_a^b} f(x) \: d \alpha (x) = A$, and let $\epsilon > 0$ be given. Then for this $\epsilon$ there exists a partition $P_{\epsilon}'$ such that if $P$ is finer than $P_{\epsilon}'$ then:
(4)
\begin{align} \quad U(P, f, \alpha) < \overline{\int_a^b} f(x) \: d \alpha (x) + \epsilon = A + \epsilon \quad (*) \end{align}
• Similarly, for this $\epsilon$ there exists a partition $P_{\epsilon}''$ such that if $P$ is finer than $P_{\epsilon}''$ we have that:
(5)
\begin{align} \quad A - \epsilon = \underline{\int_a^b} f(x) \: d \alpha( x) - \epsilon < L(P, f, \alpha) \quad (**) \end{align}
• Let $P_{\epsilon} = P_{\epsilon}' \cup P_{\epsilon}''$. Then for $P$ finer than $P_{\epsilon}$ we will have that both $(*)$ and $(**)$ hold and so:
(6)
\begin{align} \quad A - \epsilon < L(P, f, \alpha) \leq S(P, f, \alpha) \leq U(P, f, \alpha) < A + \epsilon \\ \quad \mid U(P, f, \alpha) - A \mid < \epsilon \end{align}
• So for all $\epsilon > 0$ there exists a $P_{\epsilon} \in \mathscr{P}[a, b]$ such that if $P$ is finer than $P_{\epsilon}$ then $\mid U(P, f, \alpha) - A \mid < \epsilon$ so $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ and more over:
(7)
\begin{align} \quad \int_a^b f(x) \: d \alpha (x) \end{align} 