Introduction to Riemann-Stieltjes Integrals Review

Table of Contents

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.

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:

(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}

(3)

\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:

(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}

(5)

\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:

(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}

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:

(9)

\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:

(10)

\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.

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