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a-algebra absolutely continuous binary bounded variation Cantor set cardinal Cauchy sequence choose closed interval closed set cluster point compact set complex numbers consider continuous function converges absolutely convex Corollary corresponding defined denoted diverges domain element equality holds equation example exists finite number following theorem function f Given a number Hence Hint inequality irrational Lebesgue integrals Lebesgue measure Lemma Let f limit linear m-measurable measurable function measurable set measure zero metric outer measure metric space monotonically increasing neighborhood nondecreasing nonempty set nonnegative norm notation obtain open interval open relative open set partial sums partition polynomial positive integer positive number PROBLEM Proof rational numbers real numbers real sequence Riemann integrable satisfying set G step function subset summable Take union vector space