Definition:Riemann-Stieltjes Integral

Definition

Let $\Bbb I = \closedint a b$ be a closed real interval.

Let $f, \alpha : \Bbb I \to \R$ be a real functions that are bounded on $\Bbb I$.

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Then, $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint a b$ if and only if:

there exists some $A \in \R$ such that, for every $\epsilon > 0$, there is a finite subdivision $P_\epsilon$ of $\Bbb I$ such that:

for every Riemann-Stieltjes sum $\map S {P, f, \alpha}$ of $f$ with respect to $\alpha$ for a subdivision $P$, where $P$ is finer than $P_\epsilon$:

$\size {\map S {P, f, \alpha} - A} < \epsilon$

The real number $A$ is called the Riemann-Stieltjes integral of $f$ with respect to $\alpha$ on $\closedint a b$, and is denoted:

$\ds A = \int_a^b f \rd \alpha = \int_a^b \map f x \rd \map \alpha x$

General Limits of Integration

The notation for the Riemann-Stieltjes integral:

$\ds A = \int_a^b f \rd \alpha = \int_a^b \map f x \rd \map \alpha x$

is defined for $a < b$.

We extend it to arbitrary $a, b \in \R$ by the following conventions.

If $a = b$, then:

$\ds \int_a^b f \rd \alpha = \int_a^b \map f x \rd \map \alpha x := 0$

If $a > b$, then:

$\ds \int_a^b f \rd \alpha = \int_a^b \map f x \rd \map \alpha x := - \int_b^a f \rd \alpha$

whenever $f$ is Riemann-Stieltjes integrable with respect to $\alpha$ on $\closedint b a$.

Also see

• Results about the Riemann-Stieltjes integral can be found here.

Source of Name

This entry was named for Bernhard Riemann and Thomas Joannes Stieltjes.

Sources

• 1974: Tom M. Apostol: Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $7$ The Riemann-Stieltjes Integral: $\S 7.3$: Definition $7.1$