facts about Riemann--Stieltjes integralFactsAboutRiemannStieltjesIntegral2009-05-08 13:16:592009-05-08 13:16:59TopicRiemann-Stieltjes integralRiemann Stieltjes integral% this is the default PlanetMath preamble. as your knowledge
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\begin{itemize}
\item If the integrator $g$ of the \PMlinkid{Riemann--Stieltjes integral}{3187}\, $\int_a^bf(x)\,dg(x)$ is the identity function, then the integral reduces to the Riemann integral $\int_a^bf(x)\,dx$.
\item If the integrand of the Riemann--Stieltjes integral is a constant function, one has
$$\int_a^bc\,dg(x) \;=\; c(g(b)-g(a)).$$
\item 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.
\item If $f$ is continuous, $g$ monotonically nondecreasing and differentiable on the interval \,$[a,\,b]$,\, then
$$\frac{d}{dx}\int_a^xf(t)\,dg(t) \;=\; f(x)g'(x) \quad \mbox{for\;\;} a < x < b.$$
\end{itemize}