Introduction to Riemann-Stieltjes Integrals Review

# Introduction to Riemann-Stieltjes Integrals Review

We will now review some of the recent material regarding Riemann-Stieltjes integrals.

• Recall from the Riemann-Stieltjes Integrals page that if $[a, b]$ is a closed interval, $f$ and $\alpha$ are functions defined on $[a, b]$, $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$, $t_k \in [x_{k-1}, x_k]$ for each $k \in \{1, 2, ..., n \}$, and $\alpha_k = \alpha(x_k) - \alpha(x_{k-1})$ then a Riemann-Stieltjes Sum with respect to the partition $P$ and the functions $f$ and $\alpha$ is:
(1)
\begin{align} \quad S(P, f, \alpha) = \sum_{k=1}^{n} f(t_k) \Delta \alpha_k \end{align}
• Furthermore, we say that the functions $f$ is Riemann-Stieltjes Integrable with respect to $\alpha$ on $[a, b]$ (denoted $f \in R(\alpha)$ on $[a, b]$) if there exists an $A \in \mathbb{R}$ such that for all $\epsilon > 0$ there exists a partition $P_{\epsilon} \in \mathscr{P}[a, b]$ such that for all partitions $P$ finer than $P_{\epsilon}$ ($P_{\epsilon} \subseteq P$) and for any choices of $t_k \in [x_{k-1}, x_k]$ we have that $\mid S(P, f, \alpha) - A \mid < \epsilon$.
• If such an $A \in \mathbb{R}$ exists then we we write $\displaystyle{\int_a^b f(x) \: d \alpha(x) = A}$ where we call $A$ the value of the Riemann-Stieltjes Integral.
(2)
\begin{align} \quad \int_a^b [f(x) + g(x)] \: d \alpha (x) = \int_a^b f(x) \: d \alpha (x) + \int_a^b g(x) \: d \alpha (x) \end{align}
• Furthermore, if $c \in \mathbb{R}$ then:
(3)
\begin{align} \quad \int_a^b cf(x) \: d \alpha(x) = c \int_a^b f(x) \: d \alpha (x) \end{align}
(4)
\begin{align} \quad \int_a^b f(x) \: d (\alpha + \beta)(x) = \int_a^b f(x) \: d \alpha (x) + \int_a^b f(x) \: d \beta (x) \end{align}
• Similarly, if $c \in \mathbb{R}$ then:
(5)
\begin{align} \quad \int_a^b f(x) \: d(c \alpha)(x) = c \int_a^b f(x) \: d \alpha (x) \end{align}
(6)
\begin{align} \quad \int_a^b f(x) \: d \alpha (x) = \int_a^c f(x) \: d \alpha (x) + \int_c^b f(x) \: d \alpha (x) \end{align}
• On The Formula for Integration by Parts of Riemann-Stieltjes Integrals page we saw an extremely important theorem giving us a formula called the Formula for Integration by Parts of Riemann-Stieltjes integrals. We saw that if $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $[a, b]$ then also $\alpha$ is Riemann-Stieltjes integrable with respect to $f$ on $[a, b]$ and:
(7)
\begin{align} \quad \int_a^b f(x) \: d \alpha (x) + \int_a^b \alpha (x) \: d f(x) = f(b)\alpha(b) - f(a)\alpha(a) \end{align}
• We also saw another extremely important theorem which allowed us to reduce Riemann-Stieltjes integrals to Riemann integrals under certain conditions. On the Reducing Riemann-Stieltjes Integrals to Riemann Integrals page we saw that if $f$ is Riemann-Stieltjes integrable, $f$ is bounded on $[a, b]$, $\alpha'$ exists, and $\alpha'$ is continuous on $[a, b]$ then:
(8)
\begin{align} \quad \int_a^b f(x) \: d \alpha (x) = \int_a^b f(x) \alpha' (x) \: dx \end{align}
(9)
\begin{align} \quad \int_a^b C \: d \alpha(x) = C[\alpha(b) - \alpha(a)] \end{align}
(10)
\begin{align} \quad \int_a^b f(x) \: dC = 0 \end{align}