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Riemann Integrable

  • Definition: Let P = {x_0, x_1, ... , x_n} be a partition of [a, b] and let t_k be a point in the subinterval [x_(k-1), x_k]. A sum of the form S(P, f, g) = sum(f(t_k)*(delta g_k)) is called a Riemann-Stieltjes sum of f with respect to a. We say f is Riemann-integrable with respect to g on [a, b], and we write "f is an element of R(g) on [a, b]" if there exists a number A having the following property: For every epsilon > 0, there exists a partition P_epsilon of {a, b} such that for every partition P finer than P_epsilon and for every choice of the points t_k in [x_(k-1), x_k], we have |S(P, f, a) - A| < epsilon.

    Source: Mathematical Analysis, second edition by Tom M. Apostol

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