High Quality Content by WIKIPEDIA articles! In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The Riemann–Stieltjes integral of a real-valued function of a real variable with respect to a real function g is denoted by int_a^b f(x) , dg(x) and defined to be the limit, as the mesh of the partition P={ a = x_0 < x_1 < cdots < x_n = b} of the interval [a, b] approaches zero, of the approximating sum S(P,f,g) = sum_{i=0}^{n-1} f(c_i)(g(x_{i+1})-g(x_i)) where ci is in the i-th subinterval [xi, xi+1]. The two functions and g are respectively called the integrand and the integrator. The "limit" is here understood to be a number A (the value of the Riemann-Stieltjes integral) such that for every ? > 0, there exists ? > 0 such that for every partition P with mesh(P) < ?, and for every choice of points ci in [xi, xi+1], |S(P,f,g)-A| < varepsilon. ,

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