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:

\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**.

- On the
**Linearity of the Integrand of Riemann-Stieltjes Integrals**page we proved the linearity of the integrand of Riemann-Stieltjes integrals. If $f$ and $g$ are functions defined on $[a, b]$ then:

\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:

\begin{align} \quad \int_a^b cf(x) \: d \alpha(x) = c \int_a^b f(x) \: d \alpha (x) \end{align}

- On the
**Linearity of the Integrator of Riemann-Stieltjes Integrals**page we also proved the linear of the integrator of Riemann-Stieltjes integrals. If $\alpha$ and $\beta$ are functions defined on $[a, b]$ then:

\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:

\begin{align} \quad \int_a^b f(x) \: d(c \alpha)(x) = c \int_a^b f(x) \: d \alpha (x) \end{align}

- On the
**Riemann-Stieltjes Integrability on Subintervals**page we saw that if $c \in (a, b)$ then if two of the three integrals below are known to exist then the third is guaranteed to exist:

\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:

\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:

\begin{align} \quad \int_a^b f(x) \: d \alpha (x) = \int_a^b f(x) \alpha' (x) \: dx \end{align}

- On the
**Riemann-Stieltjes Integrals with Constant Integrands**page we saw that if $f(x) = C$ for some $C \in \mathbb{R}$ and $\alpha$ is any function defined on $[a, b]$ then $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ and:

\begin{align} \quad \int_a^b C \: d \alpha(x) = C[\alpha(b) - \alpha(a)] \end{align}

- Similarly, on the
**Riemann-Stieltjes Integrals with Constant Integrators*** page we saw that if $f$ is any function defined on $[a, b]$ then $\alpha(x) = C$ for some $C \in \mathbb{R}$ then $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ and:

\begin{align} \quad \int_a^b f(x) \: dC = 0 \end{align}

- On the
**Evaluating Riemann-Stieltjes Integrals**page we looked at some basic examples of evaluating Riemann-Stieltjes integrals.