PlanetMath: Riemann-Stieltjes integral

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Riemann-Stieltjes integral

(Definition)

Let and $\alpha$ be bounded, real-valued functions defined upon a closed finite interval of $\mathbb{R} (a \neq b)$ , $P = \{ x_{0}, ..., x_{n} \}$ a partition of , and $t_{i}$ a point of the subinterval $[ x_{i - 1}, x_{i} ]$ . A sum of the form

$\displaystyle S(P, f, \alpha) = \sum_{i = 1}^{n} f(t_{i}) (\alpha(x_{i}) - \alpha(x_{i - 1}))$

is called a Riemann-Stieltjes sum of with respect to $\alpha$ . is said to be Riemann integrable with respect to $\alpha$ on if there exists $A \in \mathbb{R}$ such that given any $\epsilon > 0$ there exists a partition $P_{\epsilon}$ of for which, for all finer than $P_{\epsilon}$ and for every choice of points $t_{i}$ , we have

$\displaystyle \vert S(P, f, \alpha) - A\vert < \epsilon$

If such an exists, then it is unique and is known as the Riemann-Stieltjes integral of with respect to $\alpha$ . is known as the integrand and $\alpha$ the integrator. The integral is denoted by

$\displaystyle \int_{a}^{b}fd\alpha \quad \textrm{or} \quad \int_{a}^{b}f(x)d\alpha(x)$

"Riemann-Stieltjes integral" is owned by Mathprof. [ full author list (2) | owner history (1) ]

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