Linearity of the Integrand of Riemann-Stieltjes Integrals

Recall from the Riemann-Stieltjes Integrals page that if $[a, b]$ is a closed interval and $f, \alpha$ are functions on $[a, b]$, then $f$ is said to be Riemann-Stieltjes integral with respect to $\alpha$ on $[a, b]$ if there exists an $A \in \mathbb{R}$ such that for all $\epsilon > 0$ such that for all partitions $P \in \mathscr{P}[a, b]$ finer than $P_{\epsilon} \in \mathscr{P}[a, b]$ ($P_{\epsilon} \subseteq P$) we have that:

(1)

\begin{align} \quad \mid S(P, f, \alpha) - A \mid < \epsilon \end{align}

Where $\displaystyle{S(P, f, \alpha) = \sum_{k=1}^{n} f(t_k) \Delta \alpha_k}$ is a Riemann-Stieltjes sum for the partition $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$, $t_k \in [x_{k-1}, x_k]$ for all $k \in \{1, 2, ..., n \}$, and $\Delta \alpha_k = \alpha(x_k) - \alpha(x_{k-1})$.

If such an $A \in \mathbb{R}$ exists, we write $\int_a^b f(x) \: d\alpha(x) = A$.

We will now look at some nice linearity properties of the integrand of Riemann-Stieltjes integrals.

Theorem 1: Let $f$ and $g$ be Riemann-Stieltjes integrable functions with respect to $\alpha$ on the interval $[a, b]$ where $\int_a^b f(x) \: d \alpha(x) = A$ and $\int_a^b g(x) \: d \alpha (x) = B$. Then $f + g$ is a Riemann-Stieltjes integrable function with respect to $\alpha$ on $[a, b]$ and $\int_a^b [f(x) + g(x)] \: d \alpha(x) = A + B$.

• Proof: Let $\epsilon > 0$ b given and let $f$ and $g$ be Riemann-Stieltjes integrable functions with respect to $\alpha$ on the interval $[a, b]$.

• Since $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on the interval $[a, b]$, we have that there exists an $A \in \mathbb{R}$ such that for $\epsilon_1 = \frac{\epsilon}{2} > 0$ we have that for all partitions $P \in \mathscr{P}[a, b]$ finer than $P_{\epsilon_1}$ ($P_{\epsilon_1} \subseteq P$) that then:

(2)

\begin{align} \quad \mid S(P, f, \alpha) - A \mid < \epsilon_1 \end{align}

• Similarly, since $g$ is Riemann-Stieltjes integrable with respect to $\alpha$ on the interval $[a, b]$ we have that there exists a $B \in \mathbb{R}$ such that for $\epsilon_2 = \frac{\epsilon}{2} > 0$ we have that for all partitions $P \in \mathscr{P}[a, b]$ finer than $P_{\epsilon_2}$ ($P_{\epsilon_2} \subseteq P$) that then:

(3)

\begin{align} \quad \mid S(P, g, \alpha) - B \mid < \epsilon_2 \end{align}

• Then for all partitions $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ finer than $P_{\epsilon} = P_{\epsilon_1} \cup P_{\epsilon_2}$ and for $t_k \in [x_{k-1}, x_k]$ for each $k \in \{1, 2, ..., n \}$ that

(4)

\begin{align} \quad \mid S(P, f + g, \alpha) -(A + B) \mid &= \biggr \lvert \sum_{k=1}^{n} (f + g)(t_k) \Delta \alpha_k - (A + B) \biggr \rvert \\ \quad &= \biggr \lvert \sum_{k=1}^{n} f(t_k) \Delta \alpha_k - A + \sum_{k=1}^{n} g(t_k) \Delta \alpha_k - B \biggr \rvert \\ \quad & \leq \biggr \lvert \sum_{k=1}^{n} f(t_k) \Delta \alpha_k \biggr \rvert + \biggr \lvert \sum_{k=1}^{n} g(t_k) \Delta \alpha_k - B \biggr \rvert \\ \quad & < \epsilon_1 + \epsilon_2 = \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \end{align}

• Therefore $f + g$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ and $\int_a^b [f(x) + g(x)] \: d \alpha(x) = A + B$. $\blacksquare$

Theorem 2: Let $f$ be Riemann-Stieltjes integrable functions with respect to $\alpha$ on the interval $[a, b]$ with $\int_a^b f(x) \: d \alpha(x) = A$ and let $c \in \mathbb{R}$. Then $cf$ is a Riemann-Stieltjes integrable function with respect to $\alpha$ on $[a, b]$ and $\int_a^b cf(x) \: d \alpha(x) = cA$.

• Let $\epsilon > 0$ be given and let $f$ be a Riemann-Stieltjes integrable function with respect to $\alpha$ on the interval $[a, b]$, and let $c \in \mathbb{R}$.

• Assume that $c \neq 0$. Since $f$ is a Riemann-Stieltjes integrable function with respect to $\alpha$ on $[a, b]$ we have that there exists an $A \in \mathbb{R}$ and $\epsilon_1 = \frac{\epsilon}{\mid c \mid}$ we have that for all partitions $P \in \mathscr{P}[a, b]$ that are finer than $P_{\epsilon_1}$ ( $P_{\epsilon_1} \subset P$ ) that then:

(5)

\begin{align} \quad \mid S(P, f, \alpha) - A \mid < \epsilon_1 = \frac{\epsilon}{\mid c \mid} \end{align}

• Then for all partitions $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ finer than $P_{\epsilon} = P_{\epsilon_1}$ and for $t_k \in [x_{k-1}, x_k]$ for each $k \in \{1, 2, ..., n \}$ that:

(6)

\begin{align} \quad \mid S(P, cf, \alpha) - cA \mid = \biggr \lvert \sum_{k=1}^{n} cf(t_k) \Delta \alpha_k - cA \biggr \rvert \\ \quad &= \biggr \lvert c \left [ \sum_{k=1}^{n} f(t_k) \Delta \alpha_k - A \right ] \biggr \rvert \\ \quad & = \mid c \mid \biggr \rvert \sum_{k=1}^{n} f(t_k) \Delta \alpha_k - A \biggr \rvert \\ \quad & < \mid c \mid \epsilon_1 = \mid c \mid \cdot \frac{\epsilon}{\mid c \mid} = \epsilon \end{align}

• So $\int_a^b c f(x) \: d \alpha = cA$.

• Now suppose that $c = 0$. Then for any partition $P \in \mathscr{P}[a, b]$ we have that

(7)

\begin{align} \quad \mid S(P, 0, \alpha) - 0 \mid = \biggr \lvert \sum_{k=1}^{n} 0f(t_k) \Delta \alpha_k - 0 \biggr \rvert = 0 < \epsilon \end{align}

• So $\int_a^b 0f(x) \: d \alpha(x) = 0$.

• Therefore $cf$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ and $\int_a^b cf(x) \: d\alpha(x) = cA$. $\blacksquare$