# Upper and Lower Riemann-Stieltjes Integrals

Recall from the Upper and Lower Riemann-Stieltjes Sums page that if $f$ is a function and the integrator $\alpha$ is an increasing function, both of which are defined on $[a, b]$, and $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$, then for each $k \in \{1, 2, ..., n \}$ we define:

(1)An upper Riemann-Stieltjes sum of $f$ with respect to $\alpha$ corresponding to the partition $P$ is:

(2)Furthermore, a lower Riemann-Stieltjes sum of $f$ with respect to $\alpha$ corresponding to the partition $P$ is:

(3)On the Properties of Upper and Lower Riemann-Stieltjes Sums page we noted that if $P' \in \mathscr{P}[a, b]$ is finer than $P$ then:

(4)We also saw that for any $P_1, P_2 \in \mathscr{P}[a, b]$ that:

(5)So for finer and finer partitions $P$, the upper Riemann-Stieltjes sums get smaller while the lower Riemann-Stieltjes sums get larger, however, the set of all upper Riemann-Stieltjes sums is bounded below by any lower Riemann-Stieltjes sum, and similarly, the set of all lower Riemann-Stieltjes sums is bounded above by any upper Riemann-Stieltjes sums. We now define two special types of Riemann-Stieltjes integrals.

Definition: Let $f$ be a function on the interval $[a, b]$ and let $\alpha$ be an increasing function on $[a, b]$. The Upper Riemann-Stieltjes Integral of $f$ with respect to $\alpha$ is defined as $\displaystyle{\overline{\int_a^b} f(x) \: d \alpha(x) = \inf \{ U(P, f, \alpha) : P \in \mathscr{P}[a, b] \}}$ and the Lower Riemann-Stieltjes Integral of $f$ with respect to $\alpha$ is defined as $\displaystyle{\underline{\int_a^b} f(x) \: d \alpha(x) = \sup \{ L(P, f, \alpha) : P \in \mathscr{P}[a, b] \}}$. |

*Of course the abbreviated notation $\overline{\int_a^b} f \: d \alpha$ for the upper Riemann-Stieltjes integrals and $\underline{\int_a^b} f \: d \alpha$ for the lower Riemann-Stieltjes integrals can be used.*

Theorem 1: Let $f$ be a function defined on $[a, b]$ and let $\alpha$ be an increasing function on $[a, b]$. Then $\displaystyle{\underline{\int_a^b} f(x) \: d \alpha (x) \leq \overline{\int_a^b} f(x) \: d \alpha (x)}$. |

**Proof:**Let $P_1, P_2, \in \mathscr{P}[a, b]$. Then:

- If $P'$ is a refinement of $P_1 \cup P_2$ then:

- Since the lower Riemann-Stieltjes sums are always less than or equal to the upper Riemann-Stieltjes sums, we have that as $P$ gets finer than $P'$ that then by taking the supremum of the lower sums and the infimum of the upper sums and we get that: