Definition:Riemann-Stieltjes Sum

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 $\Bbb I$.

Let $P = \set {x_0, x_1, \dotsc, x_n}$ be a finite subdivision of $\closedint a b$.

For each $k \in \set {1, \dotsc, n}$, let $t_k \in \closedint {x_{k - 1} } {x_k}$.

Then, the summation:

$\ds \map S {P, f, \alpha} = \sum_{k \mathop = 1}^n \map f {t_k} \paren {\map \alpha {x_k} - \map \alpha {x_{k - 1} } }$

is a Riemann-Stieltjes sum of $f$ with respect to $\alpha$ for the subdivision $P$.

Also see

• Definition:Riemann Sum, for the case $\map \alpha x = x$

• Results about Riemann-Stieltjes sums 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.) ... (next): Chapter $7$ The Riemann-Stieltjes Integral: $\S 7.3$: Definition $7.1$