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facts about Riemann--Stieltjes integral
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- If the integrator $g$ of the Riemann-Stieltjes integral $\int_a^bf(x)\,dg(x)$ is the identity function, then the integral reduces to the Riemann integral $\int_a^bf(x)\,dx$ .
- If the integrand of the Riemann-Stieltjes integral is a constant function, one has $$\int_a^bc\,dg(x) \;=\; c(g(b)-g(a)).$$
- If the integrand $f$ is continuous and the integrator $g$ monotonically nondecreasing on the interval $[a,\,b]$ , then there exists a number $\xi$ on the interval such that $$\int_a^bf(x)\,dg(x) \;=\; f(\xi)(g(b)-g(a)).$$ Cf. the integral mean value theorem.
- If $f$ is continuous, $g$ monotonically nondecreasing and differentiable on the interval $[a,\,b]$ , then $$\frac{d}{dx}\int_a^xf(t)\,dg(t) \;=\; f(x)g'(x) \quad \mbox{for\;\;} a < x < b.$$
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Cross-references: differentiable, integral mean value theorem, number, interval, monotonically nondecreasing, continuous, constant function, integrand, Riemann integral, integral, identity function, integrator
This is version 1 of facts about Riemann--Stieltjes integral, born on 2009-05-08.
Object id is 11767, canonical name is FactsAboutRiemannStieltjesIntegral.
Accessed 528 times total.
Classification:
| AMS MSC: | 26A42 (Real functions :: Functions of one variable :: Integrals of Riemann, Stieltjes and Lebesgue type) |
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