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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 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. ,