definition and examples of riemann-stieltjes integral property of integral,
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The Riemann–Stieltjes integral of a real-valued function {\displaystyle f} of a real variable with respect to a real function {\displaystyle g} is denoted by
{\displaystyle \int _{a}^{b}f(x)\,dg(x)}
and defined to be the limit, as the norm (or mesh) of the partition (i.e. the length of the longest subinterval)
{\displaystyle P=\{a=x_{0}<x_{1}<\cdots <x_{n}=b\}}
of the interval [a, b] approaches zero, of the approximating sum
{\displaystyle S(P,f,g)=\sum _{i=0}^{n-1}f(c_{i})(g(x_{i+1})-g(x_{i}))}
where {\displaystyle c_{i}} is in the i-th subinterval [xi, xi+1]. The two functions {\displaystyle f} and {\displaystyle g} are respectively called the integrand and the integrator. Typically {\displaystyle g} is taken to be monotone (or at least of bounded variation) and right-semicontinuous (however this last is essentially convention). We specifically do not require {\displaystyle g} to be continuous, which allows for integrals that have point mass terms.
The "limit" is here understood to be a number A (the value of the Riemann–Stieltjes integral) such that for every ε > 0, there exists δ > 0 such that for every partition P with mesh(P) < δ, and for every choice of points ci in [xi, xi+1],
{\displaystyle |S(P,f,g)-A|<\varepsilon .\,}
Generalized Riemann–Stieltjes integral
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A slight generalization, introduced by Pollard (1920) and now standard in analysis, is to consider in the above definition partitions P that refine another partition Pε, meaning that P arises from Pε by the addition of points, rather than from partitions with a finer mesh. Specifically, the generalized Riemann–Stieltjes integral of f with respect to g is a number A such that for every ε > 0 there exists a partition Pε such that for every partition P that refines Pε,
{\displaystyle |S(P,f,g)-A|<\varepsilon \,}
for every choice of points ci in [xi, xi+1].
This generalization exhibits the Riemann–Stieltjes integral as the Moore–Smith limit on the directed set of partitions of [a, b] (McShane 1952). Hildebrandt (1938) calls it the Pollard–Moore–Stieltjes integral.
Darboux sums
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The Riemann–Stieltjes integral can be efficiently handled using an appropriate generalization of Darboux sums. For a partition P and a nondecreasing function g on [a, b] define the upper Darboux sum of f with respect to g by
{\displaystyle U(P,f,g)=\sum _{i=1}^{n}\,\,[\,g(x_{i})-g(x_{i-1})\,]\,\sup _{x\in [x_{i-1},x_{i}]}f(x)}
and the lower sum by
{\displaystyle L(P,f,g)=\sum _{i=1}^{n}\,\,[\,g(x_{i})-g(x_{i-1})\,]\,\inf _{x\in [x_{i-1},x_{i}]}f(x)} .
Then the generalized Riemann–Stieltjes of f with respect to g exists if and only if, for every ε > 0, there exists a partition P such that
{\displaystyle U(P,f,g)-L(P,f,g)<\varepsilon .}
Furthermore, f is Riemann–Stieltjes integrable with respect to g (in the classical sense) if
{\displaystyle \lim _{\operatorname {mesh} (P)\to 0}[\,U(P,f,g)-L(P,f,g)\,]=0.}
See Graves (1946, Chap. XII, §3).
Examples
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Riemann Integral
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Of course, the standard Riemann integral is an example of the Riemann-Stieltjes integral if {\displaystyle g(x)} is taken as {\displaystyle x}.
Differentiable {\displaystyle g(x)}
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Given a {\displaystyle g(x)} which is continuously differentiable over {\displaystyle \mathbb {R} } it can be shown that there is the equality
{\displaystyle \int _{a}^{b}f(x)dg(x)=\int _{a}^{b}f(x)g'(x)dx}
where the integral on the right-hand side is the standard Riemann-integral. This of course assumes that {\displaystyle f} is already integrable for this Riemann-Stieltjes integral. More generally, the Riemann integral equals the Riemann–Stieltjes integrals if {\displaystyle g} is the (Lebesgue) integral of its derivative; in this case {\displaystyle g} is said to be absolutely continuous. It may be the case that {\displaystyle g} has jump discontinuities, or may have derivative zero almost everywhere while still being continuous and increasing (for example, {\displaystyle g} could be the Cantor function or De