Upper and Lower Riemann-Stieltjes Integrals of Sums of Integrands

# Upper and Lower Riemann-Stieltjes Integrals of Sums of Integrands

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}

Now suppose that we have two functions $f$ and $g$ with an increasing function $\alpha$. What can be said about the upper and lower Riemann-Stieltjes integrals of $f + g$ with respect to $\alpha$? Unfortunately equality does not always hold as we see in the following theorem.

 Theorem 1: Let $f$ and $g$ be any functions defined on $[a, b]$ and let $\alpha$ be an increasing function on $[a, b]$. Then: a) $\displaystyle{\overline{\int_a^b} [f(x) + g(x)] \: d \alpha (x) \leq \overline{\int_a^b} f(x) \: d \alpha (x) + \overline{\int_a^b} g(x) \: d \alpha (x)}$. b) $\displaystyle{\underline{\int_a^b} [f(x) + g(x)] \: d \alpha (x) \geq \underline{\int_a^b} f(x) \: d \alpha(x) + \underline{\int_a^b} g(x) \: d \alpha (x)}$.
• Proof of a) Let $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$. Note that if $f, g : A \to \mathbb{R}$ are functions, then $f(x) \leq \sup \{ f(x) : x \in A \}$ and $g(x) \leq \sup \{ g(x) : x \in A \}$. Therefore:
(3)
\begin{align} \quad f(x) + g(x) \leq \sup \{ f(x) : x \in A \} + \sup \{ g(x) : x \in A \} \end{align}
• So $\sup \{ f(x) : x \in A \} + \sup \{ g(x) : x \in A \}$ is an upper bound to $f(x) + g(x)$ on $[a, b]$. Hence:
(4)
\begin{align} \quad \sup \{ f(x) + g(x) : x \in [a, b] \} \leq \sup \{ f(x) : x \in A \} + \sup \{ g(x) : x \in A \} \: (*) \end{align}
• Using this we have that:
(5)
\begin{align} \quad \overline{\int_a^b} [f(x) + g(x)] \: d \alpha (x) &= \inf \{ U(P, f + g, \alpha) : P \in \mathscr{P}[a, b] \} \\ \quad &= \inf \left \{ \sum_{k=1}^{n} M_k(f + g) \Delta \alpha_k : P \in \mathscr{P}[a, b] \right \} \\ \quad &= \inf \left \{ \sum_{k=1}^{n} \sup \{ f(x) + g(x) : x \in [x_{k-1}, x_k] \} \Delta \alpha_k : P \in \mathscr{P}[a, b] \right \} \\ \quad & \overset {(*)} \leq \inf \left \{ \sum_{k=1}^{n} [\sup \{ f(x) : x \in [x_{k-1}, x_k] \} + \sup \{ g(x) : x \in [x_{k-1}, x_k] \}] \Delta \alpha_k : P \in \mathscr{P}[a, b] \right \} \\ \quad & \leq \inf \left \{ \sum_{k=1}^{n} \sup \{ f(x) : x \in [x_{k-1}, x_k] \} \Delta \alpha_k : P \in \mathscr{P}[a, b] \right \} + \inf \left \{ \sum_{k=1}^{n} \sup \{ g(x) : x \in [x_{k-1}, x_k] \} \Delta \alpha_k : P \in \mathscr{P}[a, b] \right \} \\ \quad & \leq \overline{\int_a^b} f(x) \: d \alpha (x) + \overline{\int_a^b} g(x) \: d \alpha (x) \quad \blacksquare \end{align}
• Proof of b) Let $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$. Note that if $f, g : A \to \mathbb{R}$ are functions, then $f(x) \geq \inf \{ f(x) : x \in A \}$ and $g(x) \geq \inf \{ g(x) : x \in A \}$. Therefore:
(6)
\begin{align} \quad f(x) + g(x) \geq \inf \{ f(x) : x \in A \} + \inf \{ g(x) : x \in A \} \end{align}
• So $\inf \{ f(x) : x \in A \} + \inf \{ g(x) : x \in A \}$ is a lower bound to $f(x) + g(x)$ on $[a, b]$. Hence:
(7)
\begin{align} \quad \inf \{ f(x) + g(x) : x \in [a, b] \} \geq \inf \{ f(x) : x \in A \} + \inf \{ g(x) : x \in A \} \: (**) \end{align}
• Using this and we have that:
(8)
\begin{align} \quad \underline{\int_a^b} [f(x) + g(x)] \: d \alpha (x) &= \sup \{ L(P, f + g, \alpha) : P \in \mathscr{P}[a, b] \} \\ \quad &= \sup \left \{ \sum_{k=1}^{n} m_k(f + g) \Delta \alpha_k : P \in \mathscr{P}[a, b] \right \} \\ \quad &= \sup \left \{ \sum_{k=1}^{n} \inf \{ f(x) + g(x) : x \in [x_{k-1}, x_k] \} \Delta \alpha_k : P \in \mathscr{P}[a, b] \right \} \\ \quad & \overset{(**)} \geq \sup \left \{ \sum_{k=1}^{n} [\inf \{ f(x) : x \in [x_{k-1}, x_k] \} + \inf \{ g(x) : x \in [x_{k-1}, x_k] \}] \Delta \alpha_k : P \in \mathscr{P}[a, b] \right \} \\ \quad & \geq \sup \left \{ \sum_{k=1}^{n} \inf \{ f(x) : x \in [x_{k-1}, x_k] \} \Delta \alpha_k : P \in \mathscr{P}[a, b] \right \} + \sup \left \{ \sum_{k=1}^{n} \inf \{ g(x) : x \in [x_{k-1}, x_k] \} \Delta \alpha_k : P \in \mathscr{P}[a, b] \right \} \\ \quad & \geq \underline{\int_a^b} f(x) \: d \alpha (x) + \underline{\int_a^b} g(x) \: d \alpha (x) \quad \blacksquare \end{align}