Upper and Lower Riemann-Stieltjes Sums

We will now look more into the theory of Riemann-Stieltjes integrals by assuming that the the integrator $\alpha$ is increasing on the interval $[a, b]$. Then for all $x, y \in [a, b]$ such that $x \leq y$ we will have that $\alpha(x) < \alpha(y)$ and so $\alpha(y) - \alpha(x) \geq 0$.

When the integrator $\alpha$ is increasing on $[a, b]$, we have that for all partitions $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ that the term $\Delta \alpha_k = \alpha (x_k) - \alpha(x_{k-1}) \geq 0$ is the Riemann-Stieltjes sum $S(P, f, \alpha)$ is nonnegative. We will see the importance of $\Delta \alpha_k \geq 0$ shortly. We will first define two special types of Riemann-Stieltjes sums.

Definition: Let $f$ and $\alpha$ be functions defined on the interval $[a, b]$ and let $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$. Let $M_k(f) = \sup \{ f(x) : x \in [x_{k-1}, x_k] \}$ and let $m_k (f) = \inf \{ f(x) : x \in [x_{k-1}, x_k] \}$. Then a Upper Riemann-Stieltjes Sum corresponding to the partition $P$ is denoted $U(P, f, \alpha)$ and equals $U(P, f, \alpha) = \sum_{k=1}^{n} M_k (f) \Delta \alpha_k$. A Lower Reimann-Stieltjes Sum corresponding to the partition $P$ is denoted $L(P, f, \alpha)$ and equals $L(P, f, \alpha) = \sum_{k=1}^{n} m_k (f) \Delta \alpha_k$.

Let $t_k \in [x_{k-1}, x_k]$ for all $k \in [1, 2, ..., n]$. Notice that then since $m_k(f) = \inf \{ f(x) : x \in [x_{k-1}, x_k] \}$ we must have for all $t_k \in [x_{k-1}, x_k]$ and for all $k \in \{1, 2, ..., n \}$ that:

(1)

\begin{align} \quad m_k(f) \leq f(t_k) \end{align}

Similarly, since $M_k (f) = \sup \{ f(x) : x \in [x_{k-1}, x_k] \}$ we must have that for all $t_k \in [x_{k-1}, x_k]$ and for all $k \in \{1, 2, ..., n \}$ that:

(2)

\begin{align} \quad f(t_k) \leq M_k (f) \end{align}

So for all $t_k \in [x_{k-1}, x_k]$ and for all $k \in \{ 1, 2, ..., n \}$ when we combine these inequalities we get:

(3)

\begin{align} \quad m_k (f) \leq f(t_k) \leq M_k (f) \quad (*) \end{align}

Now suppose that $\alpha$ is an increasing function on the interval $[a, b]$. Then $\alpha_k = \alpha(x_k) - \alpha(x_{k-1}) \geq 0$ as we mentioned earlier. As a result of $\alpha$ being an increasing function and $(*)$ holding we have that for each $k \in \{1, 2, ..., n \}$ that:

(4)

\begin{align} \quad m_k (f) \Delta \alpha_k \leq f(t_k) \Delta \alpha_k \leq M_k(f) \Delta \alpha_k \end{align}

Taking the sums as $k$ ranges from $1$ to $n$ gives us an important inequality when $\alpha$ is an increasing function on $[a, b]$:

(5)

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

We will look at some important properties of these upper and lower Riemann-Stieltjes sums for increasing $\alpha$ in the next section.