# Riemann-Stieltjes integral

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

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

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

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

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

 $\int_{a}^{b}fd\alpha\quad\textrm{or}\quad\int_{a}^{b}f(x)d\alpha(x)$
Title Riemann-Stieltjes integral RiemannStieltjesIntegral 2013-03-22 12:51:13 2013-03-22 12:51:13 Mathprof (13753) Mathprof (13753) 11 Mathprof (13753) Definition msc 26A42 RiemannSum IntegralSign Riemann-Stieltjes sum integrator