# facts about Riemann–Stieltjes integral

• If the integrator $g$ of the http://planetmath.org/node/3187Riemann–Stieltjes integral$\int_{a}^{b}f(x)\,dg(x)$ is the identity function, then the integral reduces to the Riemann integral $\int_{a}^{b}f(x)\,dx$.

• If the integrand of the Riemann–Stieltjes integral is a constant function, one has

 $\int_{a}^{b}\!c\,dg(x)\;=\;c\!\cdot\!(g(b)\!-\!g(a)).$
• 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

 $\int_{a}^{b}\!f(x)\,dg(x)\;=\;f(\xi)(g(b)\!-\!g(a)).$

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}^{x}\!f(t)\,dg(t)\;=\;f(x)g^{\prime}(x)\quad\mbox{for\;\;}% a
Title facts about Riemann–Stieltjes integral FactsAboutRiemannStieltjesIntegral 2013-03-22 18:55:03 2013-03-22 18:55:03 pahio (2872) pahio (2872) 5 pahio (2872) Topic msc 26A42 properties of Riemann–Stieltjes integral PropertiesOfRiemannStieltjesIntegral