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)

\begin{align} \quad M_k(f) = \sup \{ f(x) : x \in [x_{k-1}, x_k] \} \quad , \quad m_k(f) = \inf \{ f(x) : x \in [x_{k-1}, x_k] \} \end{align}

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

(2)

\begin{align} \quad U(P, f, \alpha) = \sum_{k=1}^{n} M_k(f) \Delta \alpha_k \end{align}

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

(3)

\begin{align} \quad L(P, f, \alpha) = \sum_{k=1}^{n} m_k(f) \Delta \alpha_k \end{align}

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)

\begin{align} \quad U(P', f, \alpha) \leq U(P, f, \alpha) \quad \mathrm{and} \quad L(P, f, \alpha) \leq L(P', f, \alpha) \end{align}

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

(5)

\begin{align} \quad L(P_1, f, \alpha) \leq U(P_2, f, \alpha) \end{align}

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:

(6)

\begin{align} \quad L(P_1, f, \alpha) \leq U(P_2, f, \alpha) \end{align}

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

(7)

\begin{align} \quad L(P_1, f, \alpha) \leq L(P', f, \alpha) \leq U(P', f, \alpha) \leq U(P_2, f, \alpha) \end{align}

• 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:

(8)

\begin{align} \quad \underline{\int_a^b} f(x) \: d \alpha (x) \leq \overline{\int_a^b} f(x) \: d \alpha (x) \quad \blacksquare \end{align}