Properties of Upper and Lower Riemann-Stieltjes Sums

Recall from the Upper and Lower Riemann-Stieltjes Sums page that if $f$ and $\alpha$ are functions on the interval $[a, b]$ and if $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ then let $M_k (f)$ and $m_k(f)$ be defined for all $k \in \{1, 2, ..., n \}$ by:

(1)

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

(2)

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

Then an upper Riemann-Stieltjes sum corresponding to the partition $P$ is:

(3)

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

And similarly, a lower Riemann-Stieltjes sum corresponding to the partition $P$ is:

(4)

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

Recall that if $\alpha$ is increasing on the interval $[a, b]$ then we have that for all partitions $P \in \mathscr{P}[a, b]$ that:

(5)

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

We will now look at some nice properties regarding upper and lower Riemann-Stieltjes sums when $\alpha$ is an increasing function.

Theorem 1: Let $f$ and $\alpha$ be functions defined on $[a, b]$ and let $\alpha$ be an increasing function on $[a, b]$. Then if $P, P' \subseteq \mathscr{P}[a, b]$ and $P \subseteq P'$ we have that: a) $U(P', f, \alpha) \leq U(P, f. \alpha)$ b) $L(P, f, \alpha) \leq L(P', f, \alpha)$.

In other words, if $\alpha$ is an increasing function then as $P$ gets finer and finer, the upper Riemann-Stieltjes sums get "smaller" while the lower Riemann-Stieltjes sums get "larger'.

We will only prove (a) since the proof of (b) is analogous.

• Proof of a) Let $P, P' \in \mathscr{P}[a, b]$ and let $P \subseteq P'$. Let $P = \{ a = x_0, x_1, ..., x_n = b \}$. It suffices to look at a refinement $P'$ of $P$ that contains only one additional point, say $P' = \{ a = x_0,x_1, ..., x_j, c, x_{j+1}, ..., x_n = b \}$. Consider an upper Riemann-Stieltjes sums of $P'$:

(6)

\begin{align} \quad U(P', f, \alpha) = \left ( \sum_{k=1}^{j} M_k (f) \Delta \alpha_k \right ) + M_{c} (f)[\alpha(c) - \alpha(x_j)] + M_{c^*} (f) [\alpha (x_{j+1}) - \alpha(c)] + \left ( \sum_{k=j+2}^n M_k (f) \Delta \alpha_k \right ) \end{align}

• Notice that the only difference between the $U(P, f, \alpha)$ and $U(P', f, \alpha)$ is that the term $M_{j+1} (f) [\alpha(x_{j+1}) - \alpha(x_j)]$ in the sum $U(P, f, \alpha)$ is replaced by $M_{c} (f)[\alpha(c) - \alpha(x_j)] + M_{c^*} (f) [\alpha (x_{j+1}) - \alpha(c)]$ in the sum $U(P, f', \alpha)$. Now note that:

(7)

\begin{align} \quad M_c = \sup \{ f(x) : x \in [x_j, c] \} \leq M_{j+1} = \sup \{ f(x) : x \in [x_j, x_{j+1}] \} \end{align}

• And similarly:

(8)

\begin{align} \quad M_{c^*} = \sup \{ f(x) : x \in [c, x_{j+1}] \} \leq M_{j+1} = \sup \{ f(x) : x \in [x_j, x_{j+1}] \} \end{align}

• Therefore we have that:

(9)

\begin{align} \quad M_c [\alpha (c) - \alpha(x_j)] + M_{c^*}[\alpha(x_{j+1}) - \alpha_c] \leq M_{j+1} [\alpha (c) - \alpha(x_j)] + M_{j+1}[\alpha(x_{j+1}) - \alpha (c)] = M_{j+1} \Delta \alpha_k \end{align}

• Hence $U(P', f, \alpha) \leq U(P, f, \alpha)$. $\blacksquare$

Theorem 2: Let $f$ and $\alpha$ be functions defined on $[a, b]$ and let $\alpha$ be an increasing function on $[a, b]$. Then for all $P_1, P_2 \in \mathscr{P}[a, b]$ we have that $L(P_1, f, \alpha) \leq U(P_2, f, \alpha)$.

• Proof: Let $P = P_1 \cup P_2$. Then $P$ is a refinement of the partitions $P_1$ and $P_2$, i.e., $P_1 \subseteq P$ and $P_2 \subseteq P$. By Theorem 1 we have that:

(10)

\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}

• The first inequality comes from having that $P_1 \subseteq P$ with Theorem 1, and the third inequality comes from having $P_2 \subseteq P$ with Theorem 1. $\blacksquare$