# Riemann's Condition Part 1 - The Existence of Riemann-Stieltjes Integrals with Increasing Integrators

Recall that if $f$ is a function defined on $[a, b]$ and $\alpha$ is an increasing function on $[a, b]$ then for any partition $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ we have that:

(1)As $P$ gets finer and $\| P \| \to 0$ we noted that $\displaystyle{L(P, f, \alpha) \to \underline{\int_a^b} f(x) \: d \alpha (x)}$ and $\displaystyle{U(P, f, \alpha) \to \overline{\int_a^b} f(x) \: d \alpha (x)}$. Since $L(P, f, \alpha) \leq S(P, f, \alpha) \leq U(P, f, \alpha)$ for all partitions $P$, we might expect a sort of "Squeeze theorem" for the existence of the Riemann-Stieltjes integral of $f$ with respect to $\alpha$ on $[a, b]$ provided that the difference $U(P, f, \alpha) - L(P, f, \alpha)$ can be made arbitrarily small. Fortunately, the extremely important Riemann's condition asserts just this.

The theorem below is rather long so we will prove that $a) \implies b)$ on this page and on the Riemann's Condition Part 2 - The Existence of Riemann-Stieltjes Integrals with Increasing Integrators page we will prove that $b) \implies c)$ and $c) \implies a)$.

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) $\displaystyle{\overline{\int_a^b} f(x) \: d \alpha(x) = \underline{\int_a^b} f(x) \: d \alpha (x)}$. |

*It is important to remember that the Theorem above is guaranteed to hold when $\alpha$ is an increasing function!*

*Note that since $L(P, f, \alpha) \leq U(P, f, \alpha)$ that then $U(P, f, \alpha) - L(P, f, \alpha) > 0$ for all partitions $P \in \mathscr{P}[a, b]$.*

*Also note that Riemann's Condition provides a link between $f$ being Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ and the upper and lower Riemann-Stieltjes integrals of $f$ with respect to $\alpha$ on $[a, b]$ equalling each other.*

**Proof of $a) \implies b)$:**Suppose that $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$. Then for some $A \in \mathbb{R}$ we have that:

- Let $\epsilon > 0$ be given. Since $\alpha$ is an increasing function we have that since $a < b$ that then either $\alpha(a) = \alpha (b)$ or $\alpha(a) < \alpha(b)$.

- First consider the case when $\alpha (a) = \alpha (b)$. Since $\alpha$ is increasing on $[a, b]$ this implies that $\alpha$ is constant on $[a, b]$ and so for any partition $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ we have that $\Delta \alpha_k = \alpha(x_k) - \alpha(x_{k-1}) = 0$ for all $k \in \{1, 2, ..., n \}$. Moreover, for any partition $P$ we have that $U(P, f, \alpha) = 0$ and $L(P, f, \alpha) = 0$, so Riemann's condition is satisfied trivially since for any partition $P$, $0 = U(P, f, \alpha) - L(P, f, \alpha) < \epsilon$.

- For the second suppose, suppose tht $\alpha (a) < \alpha (b)$. Then since $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ we have that for $\epsilon_1 = \frac{\epsilon}{3} > 0$ there exists a partition $P_{\epsilon_1} \in \mathscr{P}[a, b]$ such that if $P$ is finer than $P_{\epsilon_1}$ ($P_{\epsilon_1} \subseteq P$) and for any choices of $t_k, t_k' \in [x_{k-1}, x_k]$ we have that:

- Using some properties of the absolute value we see that:

- Therefore $\biggr \lvert \sum_{k=1}^{n} [f(t_k) - f(t_k')] \Delta \alpha_k \biggr \rvert < \frac{2\epsilon}{3}$. $(*)$

- We want to somehow introduce the difference of upper and lower Riemann-Stieltjes sums into our proof. We first note that:

- So for all $x, x' \in [x_{k-1}, x_k]$ we have from the definition of the supremum that:

- So for all $h > 0$ there exists $t_k, t_k' \in [x_{k-1}, x_k]$ such that:

- Choose $t_k, t_k' \in [x_{k-1}, x_k]$ such that for $h = \frac{\epsilon}{3[\alpha(b) - \alpha(a)]} > 0$, the inequality above holds. Let $P_{\epsilon} = P_{\epsilon_1}$. Then for $P$ finer than $P_{\epsilon}$ we have that $(*)$ holds and: