Splitting Upper and Lower Riemann-Stieltjes Integrals

# Splitting Upper and Lower Riemann-Stieltjes Integrals

Recall from the Upper and Lower Riemann-Stieltjes Integrals page that if $f$ is a function defined on $[a, b]$, $\alpha$ is an increasing function on $[a, b]$, then the upper Riemann-Stieltjes integral of $f$ with respect to $\alpha$ on $[a, b]$ is defined to be:

(1)
\begin{align} \quad \overline{\int_a^b} f(x) \: d \alpha(x) = \inf \{ U(P, f, \alpha) : P \in \mathscr{P}[a, b] \} \end{align}

Similarly, the lower Riemann-Stieltjes integral of $f$ with respect to $\alpha$ on $[a, b]$ is defined to be:

(2)
\begin{align} \quad \underline{\int_a^b} f(x) \: d \alpha(x) = \sup \{ L(P, f, \alpha) : P \in \mathscr{P}[a, b] \} \end{align}

We will now prove that if $c \in (a, b)$ then the upper and lower Riemann-Stieltjes integrals can be split into two upper/lower Riemann-Stieltjes integrals whose limits of integration are from $a$ to $c$ and then $c$ to $b$.

 Theorem 1: Let $f$ be a function defined on $[a, b]$ and let $\alpha$ be an increasing function on $[a, b]$. Then for any $c \in (a, b)$ we have that: a) $\displaystyle{\overline{\int_a^b} f(x) \: d \alpha (x) = \overline{\int_a^c} f(x) \: d \alpha(x) + \overline{\int_c^b} f(x) \: d \alpha (x)}$. b) $\displaystyle{\underline{\int_a^b} f(x) \: d \alpha (x) = \underline{\int_a^c} f(x) \: d \alpha(x) + \underline{\int_c^b} f(x) \: d \alpha (x)}$.

We make use of the results proven on The Supremum and Infimum of The Bounded Set (S+T) page in the following proofs.

• Proof of a) Consider the following upper Riemann-Stieltjes integral:
(3)
\begin{align} \quad \overline{\int_a^b} f(x) \: d \alpha (x) = \inf \{ (U, P, f) : P \in \mathscr{P}[a, b] \} \end{align}
• Consider a partition $P \in \mathscr{P}[a, b]$ such that $c \in P$. Then $P = P_1 \cup P_2$ where $P_1 = \{ a = x_0, x_1, ..., x_n = c \} \in \mathscr{P}[a, c]$ and $P_2 = \{ c = y_0, y_1, ..., y_m = b \} \in \mathscr{P}[c, b]$. Then:
(4)
\begin{align} \quad P = \{ a = x_0, x_1, ..., x_n = c = y_0, y_1, ..., y_m = b \} \end{align}
• Note that selecting $c \in P$ was ultimately insignificant since as $\| P \| \to 0$ we have that $U(P, f, \alpha) \to \overline{\int_a^b} f(x) \: d \alpha(x)$ and also $\| P_1 \|, \| P_2 \| \to 0$ so $U(P_1, f, \alpha) \to \overline{\int_a^c} f(x) \: d \alpha(x)$ and $U(P_2, f, \alpha) \to \overline{\int_a^b} f(x) \: d \alpha (x)$. Therefore:
(5)
\begin{align} \quad \overline{\int_a^b} f(x) \: d \alpha (x) &= \inf \{ U(P, f, \alpha) : P \in \mathscr{P}[a, b] \} \\ \quad &= \inf \left \{ \sum_{k=1}^{n} \sup \{ f(x) : x \in [x_{k-1}, x_k] \} \Delta \alpha_k + \sum_{k=1}^{m} \sup \{ f(y) : y \in [y_{k-1}, y_k] \} \Delta \alpha_k \right \} \\ \quad &= \inf \left \{ \sum_{k=1}^{n} \sup \{ f(x) : x \in [x_{k-1}, x_k] \} \Delta \alpha_k: P_1 \in \mathscr{P}[a, c] \right \} + \inf \left \{ \sum_{k=1}^{m} \sup \{ f(y) : y \in [y_{k-1}, y_k] \} \Delta \alpha_k : P_2 \in \mathscr{P}[c, b] \right \} \quad &= \inf \{ U(P_1, f, \alpha) + U(P_2, f, \alpha) : P_1 \in \mathscr{P}[a, c], P_2 \in \mathscr{P}[c, b] \} \\ \quad &= \inf \{ U(P_1, f, \alpha) : P_1 \in \mathscr{P}[a, c] \} + \inf \{ U(P_2, f, \alpha) : P_2 \in \mathscr{P}[c, b] \} \\ \quad &= \overline{\int_a^c} f(x) \: d \alpha(x) + \overline{\int_c^b} f(x) \: d \alpha (x) \quad \blacksquare \end{align}
• Proof of b) Similarly, consider the following lower Riemann-Stieltjes integral:
(6)
\begin{align} \quad \underline{\int_a^b} f(x) \: d \alpha (x) = \sup \{ L(P, f, \alpha) : P \in \mathscr{P}[a, b] \} \end{align}
• Consider a partition $P \in \mathscr{P}[a, b]$ such that $c \in P$. Then $P = P_1 \cap P_2$ where $P_1 = \{a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, c]$ and $P_2 = \{ b = y_0, y_1, ..., y_m = c \} \in \mathscr{P}[c, b]$. Then:
(7)
\begin{align} \quad P = \{ a = x_0, x_1, ..., x_n = c = y_0, y_1, ..., y_m = b \} \end{align}
• Therefore:
(8)
\begin{align} \quad \underline{\int_a^b} f(x) \: d \alpha (x) & = \sup \{ L(P, f, \alpha) : P \in \mathscr{P}[a, b] \} \\ \quad &= \sup \left \{ \sum_{k=1}^{n} \inf \{ f(x) : x \in [x_{k-1}, x_k] \} \Delta \alpha_k + \sum_{k=1}^{m} \inf \{ f(y) : y \in [y_{k-1}, y_k] \} \Delta \alpha_k\}\right \} \\ \quad &= \sup \left \{ \sum_{k=1}^{n} \inf \{ f(x) : x \in [x_{k-1}, x_k] \} \Delta \alpha_k : P_1 \in \mathscr{P}[a, c] \right \} + \sup \left \{ \sum_{k=1}^{m} \inf \{ f(y) : y \in [y_{k-1}, y_k] \} \Delta \alpha_k : P_2 \in \mathscr{P}[c, b] \right \} \\ \quad &= \sup \{ L(P_1, f, \alpha) : P_1 \in \mathscr{P}[a, c] \} + \sup \{ L(P_2, f, \alpha) : P_2 \in \mathscr{P}[c, b] \} \\ \quad &= \underline{\int_a^c} f(x) \: d \alpha (x) + \underline{\int_c^b} f(x) \: d \alpha (x) \quad \blacksquare \end{align}